problem solving strategies physics

4.6 Problem-Solving Strategies

Learning objectives.

By the end of this section, you will be able to:

Understand and apply a problem-solving procedure to solve problems using Newton's laws of motion.

Success in problem solving is obviously necessary to understand and apply physical principles, not to mention the more immediate need of passing exams. The basics of problem solving, presented earlier in this text, are followed here, but specific strategies useful in applying Newton’s laws of motion are emphasized. These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, and so the following techniques should reinforce skills you have already begun to develop.

Problem-Solving Strategy for Newton’s Laws of Motion

Step 1. As usual, it is first necessary to identify the physical principles involved. Once it is determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation . Such a sketch is shown in Figure 4.20 (a). Then, as in Figure 4.20 (b), use arrows to represent all forces, label them carefully, and make their lengths and directions correspond to the forces they represent (whenever sufficient information exists).

Step 2. Identify what needs to be determined and what is known or can be inferred from the problem as stated. That is, make a list of knowns and unknowns. Then carefully determine the system of interest . This decision is a crucial step, since Newton’s second law involves only external forces. Once the system of interest has been identified, it becomes possible to determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 4.20 (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated earlier in this chapter, the system of interest depends on what question we need to answer. This choice becomes easier with practice, eventually developing into an almost unconscious process. Skill in clearly defining systems will be beneficial in later chapters as well. A diagram showing the system of interest and all of the external forces is called a free-body diagram . Only forces are shown on free-body diagrams, not acceleration or velocity. We have drawn several of these in worked examples. Figure 4.20 (c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram.

Step 3. Once a free-body diagram is drawn, Newton’s second law can be applied to solve the problem . This is done in Figure 4.20 (d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then they add like scalars. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. This is done by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known.

Applying Newton’s Second Law

Before you write net force equations, it is critical to determine whether the system is accelerating in a particular direction. If the acceleration is zero in a particular direction, then the net force is zero in that direction. Similarly, if the acceleration is nonzero in a particular direction, then the net force is described by the equation: F net = ma F net = ma .

For example, if the system is accelerating in the horizontal direction, but it is not accelerating in the vertical direction, then you will have the following conclusions:

You will need this information in order to determine unknown forces acting in a system.

Step 4. As always, check the solution to see whether it is reasonable . In some cases, this is obvious. For example, it is reasonable to find that friction causes an object to slide down an incline more slowly than when no friction exists. In practice, intuition develops gradually through problem solving, and with experience it becomes progressively easier to judge whether an answer is reasonable. Another way to check your solution is to check the units. If you are solving for force and end up with units of m/s, then you have made a mistake.

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32 Problem-Solving Strategies

[latexpage]

Learning Objectives

Understand and apply a problem-solving procedure to solve problems using Newton’s laws of motion.

Problem-Solving Strategy for Newton’s Laws of Motion

Step 1. As usual, it is first necessary to identify the physical principles involved. Once it is determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation . Such a sketch is shown in (Figure) (a). Then, as in (Figure) (b), use arrows to represent all forces, label them carefully, and make their lengths and directions correspond to the forces they represent (whenever sufficient information exists).

(a) A sketch is shown of a man hanging from a vine. (b) The forces acting on the person, shown by vector arrows, are tension T, pointing upward at the hand of the man, F sub T, from the same point but in a downward direction, and weight W, acting downward from his stomach. (c) In figure (c) we define only the man as the system of interest. Tension T is acting upward from his hand. The weight W acts in a downward direction. In a free-body diagram W is shown by an arrow acting downward and T is shown by an arrow acting vertically upward. (d) Tension T is shown by an arrow vertically upward and another vector, weight W, is shown by an arrow vertically downward, both having the same lengths. It is indicated that T is equal to minus W.

Step 2. Identify what needs to be determined and what is known or can be inferred from the problem as stated. That is, make a list of knowns and unknowns. Then carefully determine the system of interest . This decision is a crucial step, since Newton’s second law involves only external forces. Once the system of interest has been identified, it becomes possible to determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See (Figure) (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated earlier in this chapter, the system of interest depends on what question we need to answer. This choice becomes easier with practice, eventually developing into an almost unconscious process. Skill in clearly defining systems will be beneficial in later chapters as well. A diagram showing the system of interest and all of the external forces is called a free-body diagram . Only forces are shown on free-body diagrams, not acceleration or velocity. We have drawn several of these in worked examples. (Figure) (c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram.

Step 3. Once a free-body diagram is drawn, Newton’s second law can be applied to solve the problem . This is done in (Figure) (d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then they add like scalars. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. This is done by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known.

For example, if the system is accelerating in the horizontal direction, but it is not accelerating in the vertical direction, then you will have the following conclusions:

You will need this information in order to determine unknown forces acting in a system.

Section Summary

Draw a sketch of the problem.
Identify known and unknown quantities, and identify the system of interest. Draw a free-body diagram, which is a sketch showing all of the forces acting on an object. The object is represented by a dot, and the forces are represented by vectors extending in different directions from the dot. If vectors act in directions that are not horizontal or vertical, resolve the vectors into horizontal and vertical components and draw them on the free-body diagram.
Write Newton’s second law in the horizontal and vertical directions and add the forces acting on the object. If the object does not accelerate in a particular direction (for example, the $x$-direction) then ${F}_{\text{net}\phantom{\rule{0.25em}{0ex}}x}=0$. If the object does accelerate in that direction, ${F}_{\text{net}\phantom{\rule{0.25em}{0ex}}x}=\text{ma}$.
Check your answer. Is the answer reasonable? Are the units correct?

Problem Exercises

A $5\text{.}\text{00}×{\text{10}}^{5}\text{-kg}$ rocket is accelerating straight up. Its engines produce $1\text{.}\text{250}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{N}$ of thrust, and air resistance is $4\text{.}\text{50}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{N}$. What is the rocket’s acceleration? Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

An object of mass m is shown. Three forces acting on it are tension T, shown by an arrow acting vertically upward, and friction f and gravity m g, shown by two arrows acting vertically downward.

Using the free-body diagram:

${F}_{\text{net}}=T-f-mg=\text{ma}$,

$a=\frac{T-f-\text{mg}}{m}=\frac{1\text{.}\text{250}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{N}-4.50×{\text{10}}^{\text{6}}\phantom{\rule{0.25em}{0ex}}N-\left(5.00×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(9.{\text{80 m/s}}^{2}\right)}{5.00×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{kg}}=\text{6.20}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$.

The wheels of a midsize car exert a force of 2100 N backward on the road to accelerate the car in the forward direction. If the force of friction including air resistance is 250 N and the acceleration of the car is $1\text{.}{\text{80 m/s}}^{2}$, what is the mass of the car plus its occupants? Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. For this situation, draw a free-body diagram and write the net force equation.

Calculate the force a 70.0-kg high jumper must exert on the ground to produce an upward acceleration 4.00 times the acceleration due to gravity. Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

Two forces are acting on an object of mass m: F, shown by an arrow pointing upward, and its weight w, shown by an arrow pointing downward. Acceleration a is represented by a vector arrow pointing upward. The figure depicts the forces acting on a high jumper.

Find: $F$.

$F=\left(\text{70.0 kg}\right)\left[\left(\text{39}\text{.}{\text{2 m/s}}^{2}\right)+\left(9\text{.}{\text{80 m/s}}^{2}\right)\right]$$=3.\text{43}×{\text{10}}^{3}\text{N}$. The force exerted by the high-jumper is actually down on the ground, but $F$ is up from the ground and makes him jump.

This result is reasonable, since it is quite possible for a person to exert a force of the magnitude of ${\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{N}$.

When landing after a spectacular somersault, a 40.0-kg gymnast decelerates by pushing straight down on the mat. Calculate the force she must exert if her deceleration is 7.00 times the acceleration due to gravity. Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

A freight train consists of two $8.00×{10}^{4}\text{-kg}$ engines and 45 cars with average masses of $5.50×{10}^{4}\phantom{\rule{0.25em}{0ex}}\text{kg}$ . (a) What force must each engine exert backward on the track to accelerate the train at a rate of $5.00×{\text{10}}^{\text{–2}}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ if the force of friction is $7\text{.}\text{50}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N}$, assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?

(a) $4\text{.}\text{41}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N}$

(b) $1\text{.}\text{50}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N}$

Commercial airplanes are sometimes pushed out of the passenger loading area by a tractor. (a) An 1800-kg tractor exerts a force of $1\text{.}\text{75}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N}$ backward on the pavement, and the system experiences forces resisting motion that total 2400 N. If the acceleration is $0\text{.}{\text{150 m/s}}^{2}$, what is the mass of the airplane? (b) Calculate the force exerted by the tractor on the airplane, assuming 2200 N of the friction is experienced by the airplane. (c) Draw two sketches showing the systems of interest used to solve each part, including the free-body diagrams for each.

A 1100-kg car pulls a boat on a trailer. (a) What total force resists the motion of the car, boat, and trailer, if the car exerts a 1900-N force on the road and produces an acceleration of $0\text{.}{\text{550 m/s}}^{2}$? The mass of the boat plus trailer is 700 kg. (b) What is the force in the hitch between the car and the trailer if 80% of the resisting forces are experienced by the boat and trailer?

(a) $\text{910 N}$

(b) $1\text{.}\text{11}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{N}$

(a) Find the magnitudes of the forces ${\mathbf{\text{F}}}_{1}$ and ${\mathbf{\text{F}}}_{2}$ that add to give the total force ${\mathbf{\text{F}}}_{\text{tot}}$ shown in (Figure) . This may be done either graphically or by using trigonometry. (b) Show graphically that the same total force is obtained independent of the order of addition of ${\mathbf{\text{F}}}_{1}$ and ${\mathbf{\text{F}}}_{2}$. (c) Find the direction and magnitude of some other pair of vectors that add to give ${\mathbf{\text{F}}}_{\text{tot}}$. Draw these to scale on the same drawing used in part (b) or a similar picture.

A right triangle is shown made up of three vectors. The first vector, F sub one, is along the triangle’s base toward the right; the second vector, F sub two, is along the perpendicular side pointing upward; and the third vector, F sub tot, is along the hypotenuse pointing up the incline. The magnitude of F sub tot is twenty newtons. In a free-body diagram, F sub one is shown by an arrow pointing right and F sub two is shown by an arrow acting vertically upward.

Two children pull a third child on a snow saucer sled exerting forces ${\mathbf{\text{F}}}_{1}$ and ${\mathbf{\text{F}}}_{2}$ as shown from above in (Figure) . Find the acceleration of the 49.00-kg sled and child system. Note that the direction of the frictional force is unspecified; it will be in the opposite direction of the sum of ${\mathbf{\text{F}}}_{1}$ and ${\mathbf{\text{F}}}_{2}$.

$a=\text{0.139 m/s}$, $\theta =12.4º$ north of east

An overhead view of a child sitting on a snow saucer sled. Two forces, F sub one equal to ten newtons and F sub two equal to eight newtons, are acting toward the right. F sub one makes an angle of forty-five degrees from the x axis and F sub two makes an angle of thirty degrees from the x axis in a clockwise direction. A friction force f is equal to seven point five newtons, shown by a vector pointing in negative x direction. In the free-body diagram, F sub one and F sub two are shown by arrows toward the right, making a forty-five degree angle above the horizontal and a thirty-degree angle below the horizontal respectively. The friction force f is shown by an arrow along the negative x axis.

Suppose your car was mired deeply in the mud and you wanted to use the method illustrated in (Figure) to pull it out. (a) What force would you have to exert perpendicular to the center of the rope to produce a force of 12,000 N on the car if the angle is 2.00°? In this part, explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. (b) Real ropes stretch under such forces. What force would be exerted on the car if the angle increases to 7.00° and you still apply the force found in part (a) to its center?

What force is exerted on the tooth in (Figure) if the tension in the wire is 25.0 N? Note that the force applied to the tooth is smaller than the tension in the wire, but this is necessitated by practical considerations of how force can be applied in the mouth. Explicitly show how you follow steps in the Problem-Solving Strategy for Newton’s laws of motion.

Use Newton’s laws since we are looking for forces.

A horizontal dotted line with two vectors extending downward from the mid-point of the dotted line, both at angles of fifteen degrees. A third vector points straight downward from the intersection of the first two angles, bisecting them; it is perpendicular to the dotted line.

The x -components of the tension cancel. $\sum {F}_{x}=0$.

This seems reasonable, since the applied tensions should be greater than the force applied to the tooth.

Cross-section of jaw with sixteen teeth is shown. Braces are along the outside of the teeth. Three forces are acting on the protruding tooth. The applied force, F sub app, is shown by an arrow vertically downward; a second force, T, is shown by an arrow making an angle of fifteen degrees below the positive x axis; and a third force, T, is shown by an arrow making an angle of fifteen degrees below the negative x axis.

(Figure) shows Superhero and Trusty Sidekick hanging motionless from a rope. Superhero’s mass is 90.0 kg, while Trusty Sidekick’s is 55.0 kg, and the mass of the rope is negligible. (a) Draw a free-body diagram of the situation showing all forces acting on Superhero, Trusty Sidekick, and the rope. (b) Find the tension in the rope above Superhero. (c) Find the tension in the rope between Superhero and Trusty Sidekick. Indicate on your free-body diagram the system of interest used to solve each part.

Two caped superheroes hang on a rope suspended vertically from a bar.

A nurse pushes a cart by exerting a force on the handle at a downward angle $\text{35.0º}$ below the horizontal. The loaded cart has a mass of 28.0 kg, and the force of friction is 60.0 N. (a) Draw a free-body diagram for the system of interest. (b) What force must the nurse exert to move at a constant velocity?

Construct Your Own Problem Consider the tension in an elevator cable during the time the elevator starts from rest and accelerates its load upward to some cruising velocity. Taking the elevator and its load to be the system of interest, draw a free-body diagram. Then calculate the tension in the cable. Among the things to consider are the mass of the elevator and its load, the final velocity, and the time taken to reach that velocity.

Construct Your Own Problem Consider two people pushing a toboggan with four children on it up a snow-covered slope. Construct a problem in which you calculate the acceleration of the toboggan and its load. Include a free-body diagram of the appropriate system of interest as the basis for your analysis. Show vector forces and their components and explain the choice of coordinates. Among the things to be considered are the forces exerted by those pushing, the angle of the slope, and the masses of the toboggan and children.

Unreasonable Results (a) Repeat (Figure) , but assume an acceleration of $1\text{.}{\text{20 m/s}}^{2}$ is produced. (b) What is unreasonable about the result? (c) Which premise is unreasonable, and why is it unreasonable?

Unreasonable Results (a) What is the initial acceleration of a rocket that has a mass of $1\text{.}\text{50}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{kg}$ at takeoff, the engines of which produce a thrust of $2\text{.}\text{00}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{N}$? Do not neglect gravity. (b) What is unreasonable about the result? (This result has been unintentionally achieved by several real rockets.) (c) Which premise is unreasonable, or which premises are inconsistent? (You may find it useful to compare this problem to the rocket problem earlier in this section.)

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19 4.6 Problem-Solving Strategies

Understand and apply a problem-solving procedure to solve problems using Newton’s laws of motion.

Problem-Solving Strategy for Newton’s Laws of Motion

Step 1. As usual, it is first necessary to identify the physical principles involved. Once it is determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation . Such a sketch is shown in Figure 1 (a). Then, as in Figure 1 (b), use arrows to represent all forces, label them carefully, and make their lengths and directions correspond to the forces they represent (whenever sufficient information exists).

Step 2. Identify what needs to be determined and what is known or can be inferred from the problem as stated. That is, make a list of knowns and unknowns. Then carefully determine the system of interest . This decision is a crucial step, since Newton’s second law involves only external forces. Once the system of interest has been identified, it becomes possible to determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 1 (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated earlier in this chapter, the system of interest depends on what question we need to answer. This choice becomes easier with practice, eventually developing into an almost unconscious process. Skill in clearly defining systems will be beneficial in later chapters as well.

A diagram showing the system of interest and all of the external forces is called a free-body diagram . Only forces are shown on free-body diagrams, not acceleration or velocity. We have drawn several of these in worked examples. Figure 1 (c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram.

Step 3. Once a free-body diagram is drawn, Newton’s second law can be applied to solve the problem . This is done in Figure 1 (d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then they add like scalars. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. This is done by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known.

Applying Newton’s Second Law

For example, if the system is accelerating in the horizontal direction, but it is not accelerating in the vertical direction, then you will have the following conclusions:

You will need this information in order to determine unknown forces acting in a system.

Section Summary

Draw a sketch of the problem.
Identify known and unknown quantities, and identify the system of interest. Draw a free-body diagram, which is a sketch showing all of the forces acting on an object. The object is represented by a dot, and the forces are represented by vectors extending in different directions from the dot. If vectors act in directions that are not horizontal or vertical, resolve the vectors into horizontal and vertical components and draw them on the free-body diagram.
Check your answer. Is the answer reasonable? Are the units correct?

Problems & Exercises

3: Calculate the force a 70.0-kg high jumper must exert on the ground to produce an upward acceleration 4.00 times the acceleration due to gravity. Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

4: When landing after a spectacular somersault, a 40.0-kg gymnast decelerates by pushing straight down on the mat. Calculate the force she must exert if her deceleration is 7.00 times the acceleration due to gravity. Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

10: Suppose your car was mired deeply in the mud and you wanted to use the method illustrated in Figure 4 to pull it out. (a) What force would you have to exert perpendicular to the center of the rope to produce a force of 12,000 N on the car if the angle is 2.00°? In this part, explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. (b) Real ropes stretch under such forces. What force would be exerted on the car if the angle increases to 7.00° and you still apply the force found in part (a) to its center?

11: What force is exerted on the tooth in Figure 5 if the tension in the wire is 25.0 N? Note that the force applied to the tooth is smaller than the tension in the wire, but this is necessitated by practical considerations of how force can be applied in the mouth. Explicitly show how you follow steps in the Problem-Solving Strategy for Newton’s laws of motion.

12: Figure 6 shows Superhero and Trusty Sidekick hanging motionless from a rope. Superhero’s mass is 90.0 kg, while Trusty Sidekick’s is 55.0 kg, and the mass of the rope is negligible. (a) Draw a free-body diagram of the situation showing all forces acting on Superhero, Trusty Sidekick, and the rope. (b) Find the tension in the rope above Superhero. (c) Find the tension in the rope between Superhero and Trusty Sidekick. Indicate on your free-body diagram the system of interest used to solve each part.

14: Construct Your Own Problem Consider the tension in an elevator cable during the time the elevator starts from rest and accelerates its load upward to some cruising velocity. Taking the elevator and its load to be the system of interest, draw a free-body diagram. Then calculate the tension in the cable. Among the things to consider are the mass of the elevator and its load, the final velocity, and the time taken to reach that velocity.

15: Construct Your Own Problem Consider two people pushing a toboggan with four children on it up a snow-covered slope. Construct a problem in which you calculate the acceleration of the toboggan and its load. Include a free-body diagram of the appropriate system of interest as the basis for your analysis. Show vector forces and their components and explain the choice of coordinates. Among the things to be considered are the forces exerted by those pushing, the angle of the slope, and the masses of the toboggan and children.

Using the free-body diagram:

$latex \boldsymbol{ a = \frac{T – f – mg}{m} = \frac{1.250 \times 10^7 \; \textbf{N} – 4.50 \times 10^6 \; N – (5.00 \times 10^5 \;\textbf{kg})(9.80 \;\textbf{m/s}^2)}{5.00 \times 10^5 \;\textbf{kg}} = 6.20 \;\textbf{m/s}^2} $

Use Newton’s laws of motion
Use Newton’s laws since we are looking for forces.
This seems reasonable, since the applied tensions should be greater than the force applied to the tooth.

4.6 Problem-Solving Strategies by OpenStax is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Chapter 4 Dynamics: Force and Newton’s Laws of Motion

4.6 Problem-Solving Strategies

Understand and apply a problem-solving procedure to solve problems using Newton’s laws of motion.

Problem-Solving Strategy for Newton’s Laws of Motion

Step 1. As usual, it is first necessary to identify the physical principles involved. Once it is determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation . Such a sketch is shown in Figure 1 (a). Then, as in Figure 1 (b), use arrows to represent all forces, label them carefully, and make their lengths and directions correspond to the forces they represent (whenever sufficient information exists).

Step 2. Identify what needs to be determined and what is known or can be inferred from the problem as stated. That is, make a list of knowns and unknowns. Then carefully determine the system of interest . This decision is a crucial step, since Newton’s second law involves only external forces. Once the system of interest has been identified, it becomes possible to determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 1 (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated earlier in this chapter, the system of interest depends on what question we need to answer. This choice becomes easier with practice, eventually developing into an almost unconscious process. Skill in clearly defining systems will be beneficial in later chapters as well.

Applying Newton’s Second Law

For example, if the system is accelerating in the horizontal direction, but it is not accelerating in the vertical direction, then you will have the following conclusions:

You will need this information in order to determine unknown forces acting in a system.

Section Summary

Draw a sketch of the problem.
Identify known and unknown quantities, and identify the system of interest. Draw a free-body diagram, which is a sketch showing all of the forces acting on an object. The object is represented by a dot, and the forces are represented by vectors extending in different directions from the dot. If vectors act in directions that are not horizontal or vertical, resolve the vectors into horizontal and vertical components and draw them on the free-body diagram.
Write Newton’s second law in the horizontal and vertical directions and add the forces acting on the object. If the object does not accelerate in a particular direction (for example, the [latex]{x}[/latex] -direction) then [latex]{F_{\text{net} \; x}=0}.[/latex] If the object does accelerate in that direction, [latex]{F_{\text{net} \; x}=ma}.[/latex]
Check your answer. Is the answer reasonable? Are the units correct?

Problems & Exercises

1: A [latex]{5.00\times10^5\text{-kg}}[/latex] rocket is accelerating straight up. Its engines produce [latex]{1.250\times10^7\text{ N}}[/latex] of thrust, and air resistance is [latex]{4.50\times10^6\text{ N}}.[/latex] What is the rocket’s acceleration? Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

2: The wheels of a midsize car exert a force of 2100 N backward on the road to accelerate the car in the forward direction. If the force of friction including air resistance is 250 N and the acceleration of the car is [latex]{1.80\text{ m/s}^2},[/latex] what is the mass of the car plus its occupants? Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. For this situation, draw a free-body diagram and write the net force equation.

5: A freight train consists of two [latex]{8.00\times10^4\text{-kg}}[/latex] engines and 45 cars with average masses of [latex]{5.50\times10^4\text{ kg}}.[/latex] (a) What force must each engine exert backward on the track to accelerate the train at a rate of [latex]{5.00\times10^{-2}\text{ m/s}^2}[/latex] if the force of friction is [latex]{7.50\times10^5\text{ N}},[/latex] assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?

6: Commercial airplanes are sometimes pushed out of the passenger loading area by a tractor. (a) An 1800-kg tractor exerts a force of [latex]{1.75\times10^4\text{ N}}[/latex] backward on the pavement, and the system experiences forces resisting motion that total 2400 N. If the acceleration is [latex]{0.150\text{ m/s}^2},[/latex] what is the mass of the airplane? (b) Calculate the force exerted by the tractor on the airplane, assuming 2200 N of the friction is experienced by the airplane. (c) Draw two sketches showing the systems of interest used to solve each part, including the free-body diagrams for each.

7: A 1100-kg car pulls a boat on a trailer. (a) What total force resists the motion of the car, boat, and trailer, if the car exerts a 1900-N force on the road and produces an acceleration of [latex]{0.550\text{ m/s}^2}?[/latex] The mass of the boat plus trailer is 700 kg. (b) What is the force in the hitch between the car and the trailer if 80% of the resisting forces are experienced by the boat and trailer?

8: (a) Find the magnitudes of the forces [latex]{\textbf{F}_1}[/latex] and [latex]{\textbf{F}_2}[/latex] that add to give the total force [latex]{\textbf{F}_{\text{tot}}}[/latex] shown in Figure 2 . This may be done either graphically or by using trigonometry. (b) Show graphically that the same total force is obtained independent of the order of addition of [latex]{\textbf{F}_1}[/latex] and [latex]{\textbf{F}_2}.[/latex] (c) Find the direction and magnitude of some other pair of vectors that add to give [latex]{\textbf{F}_{\text{tot}}}.[/latex] Draw these to scale on the same drawing used in part (b) or a similar picture.

9: Two children pull a third child on a snow saucer sled exerting forces [latex]{\textbf{F}_1}[/latex] and [latex]{\textbf{F}_2}[/latex] as shown from above in Figure 3 . Find the acceleration of the 49.00-kg sled and child system. Note that the direction of the frictional force is unspecified; it will be in the opposite direction of the sum of [latex]{\textbf{F}_1}[/latex] and [latex]{\textbf{F}_2}.[/latex]

13: A nurse pushes a cart by exerting a force on the handle at a downward angle [latex]{35.0^{\circ}}[/latex] below the horizontal. The loaded cart has a mass of 28.0 kg, and the force of friction is 60.0 N. (a) Draw a free-body diagram for the system of interest. (b) What force must the nurse exert to move at a constant velocity?

16: Unreasonable Results (a) Repeat Exercise 7 , but assume an acceleration of [latex]{1.20\text{ m/s}^2}[/latex] is produced. (b) What is unreasonable about the result? (c) Which premise is unreasonable, and why is it unreasonable?

17: Unreasonable Results (a) What is the initial acceleration of a rocket that has a mass of [latex]{1.50\times10^6\text{ kg}}[/latex] at takeoff, the engines of which produce a thrust of [latex]{2.00\times10^6\text{ N}}?[/latex] Do not neglect gravity. (b) What is unreasonable about the result? (This result has been unintentionally achieved by several real rockets.) (c) Which premise is unreasonable, or which premises are inconsistent? (You may find it useful to compare this problem to the rocket problem earlier in this section.)

Using the free-body diagram:

[latex]{F_{\text{net}}=T-f-mg=ma},[/latex]

[latex]{ a = \frac{T - f - mg}{m} = \frac{1.250 \times 10^7 \; \textbf{N} - 4.50 \times 10^6 \; N - (5.00 \times 10^5 \;\text{kg})(9.80 \;\text{m/s}^2)}{5.00 \times 10^5 \;\text{kg}} = 6.20 \;\text{m/s}^2}[/latex]

Use Newton’s laws of motion

Find: [latex]{F}.[/latex]

[latex]{F=(70.0\text{ kg})[(39.2\text{ m/s}^2)+(9.80\text{ m/s}^2)]=3.43\times10^3\text{ N}}.[/latex] The force exerted by the high-jumper is actually down on the ground, but [latex]{F}[/latex] is up from the ground and makes him jump.

This result is reasonable, since it is quite possible for a person to exert a force of the magnitude of [latex]{10^3\text{ N}}.[/latex]

(a) [latex]{4.41\times10^5\text{ N}}[/latex]

(b) [latex]{1.50\times10^5\text{ N}}[/latex]

(a) [latex]{910\text{ N}}[/latex]

(b) [latex]{1.11\times10^3\text{ N}}[/latex]

[latex]{a=0.139\text{ m/s}},{\theta=12.4^{\circ}}[/latex] north of east

Use Newton’s laws since we are looking for forces.

The x -components of the tension cancel. [latex]{\sum{F}_x=0}.[/latex]

This seems reasonable, since the applied tensions should be greater than the force applied to the tooth.

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1 Units and Measurement

1.7 solving problems in physics, learning objectives.

By the end of this section, you will be able to:

Describe the process for developing a problem-solving strategy.
Explain how to find the numerical solution to a problem.
Summarize the process for assessing the significance of the numerical solution to a problem.

A photograph of a student’s hand, working on a problem with an open textbook, a calculator, and an eraser.

Figure 1.13 Problem-solving skills are essential to your success in physics. (credit: “scui3asteveo”/Flickr)

Problem-solving skills are clearly essential to success in a quantitative course in physics. More important, the ability to apply broad physical principles—usually represented by equations—to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life.

As you are probably well aware, a certain amount of creativity and insight is required to solve problems. No rigid procedure works every time. Creativity and insight grow with experience. With practice, the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and then progressing to the more difficult. After you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.

Although there is no simple step-by-step method that works for every problem, the following three-stage process facilitates problem solving and makes it more meaningful. The three stages are strategy, solution, and significance. This process is used in examples throughout the book. Here, we look at each stage of the process in turn.

Strategy is the beginning stage of solving a problem. The idea is to figure out exactly what the problem is and then develop a strategy for solving it. Some general advice for this stage is as follows:

Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You often need to decide which direction is positive and note that on your sketch. When you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.
Make a list of what is given or can be inferred from the problem as stated (identify the “knowns”) . Many problems are stated very succinctly and require some inspection to determine what is known. Drawing a sketch can be very useful at this point as well. Formally identifying the knowns is of particular importance in applying physics to real-world situations. For example, the word stopped means the velocity is zero at that instant. Also, we can often take initial time and position as zero by the appropriate choice of coordinate system.
Identify exactly what needs to be determined in the problem (identify the unknowns) . In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help identify the unknowns.
Determine which physical principles can help you solve the problem . Since physical principles tend to be expressed in the form of mathematical equations, a list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all the other variables are known—so you can solve for the unknown easily. If the equation contains more than one unknown, then additional equations are needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.

The solution stage is when you do the math. Substitute the knowns (along with their units) into the appropriate equation and obtain numerical solutions complete with units . That is, do the algebra, calculus, geometry, or arithmetic necessary to find the unknown from the knowns, being sure to carry the units through the calculations. This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process.

Significance

After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is not math. Rather, in doing physics, we use mathematics as a tool to help us understand nature. So, after you obtain a numerical answer, you should always assess its significance:

Check your units. If the units of the answer are incorrect, then an error has been made and you should go back over your previous steps to find it. One way to find the mistake is to check all the equations you derived for dimensional consistency. However, be warned that correct units do not guarantee the numerical part of the answer is also correct.
Check the answer to see whether it is reasonable. Does it make sense? This step is extremely important: –the goal of physics is to describe nature accurately. To determine whether the answer is reasonable, check both its magnitude and its sign, in addition to its units. The magnitude should be consistent with a rough estimate of what it should be. It should also compare reasonably with magnitudes of other quantities of the same type. The sign usually tells you about direction and should be consistent with your prior expectations. Your judgment will improve as you solve more physics problems, and it will become possible for you to make finer judgments regarding whether nature is described adequately by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to solve a problem mechanically.
Check to see whether the answer tells you something interesting. What does it mean? This is the flip side of the question: Does it make sense? Ultimately, physics is about understanding nature, and we solve physics problems to learn a little something about how nature operates. Therefore, assuming the answer does make sense, you should always take a moment to see if it tells you something about the world that you find interesting. Even if the answer to this particular problem is not very interesting to you, what about the method you used to solve it? Could the method be adapted to answer a question that you do find interesting? In many ways, it is in answering questions such as these that science progresses.

The three stages of the process for solving physics problems used in this book are as follows:

Strategy : Determine which physical principles are involved and develop a strategy for using them to solve the problem.
Solution : Do the math necessary to obtain a numerical solution complete with units.
Significance : Check the solution to make sure it makes sense (correct units, reasonable magnitude and sign) and assess its significance.

Conceptual Questions

What information do you need to choose which equation or equations to use to solve a problem?

What should you do after obtaining a numerical answer when solving a problem?

Check to make sure it makes sense and assess its significance.

Additional Problems

Consider the equation y = mt +b , where the dimension of y is length and the dimension of t is time, and m and b are constants. What are the dimensions and SI units of (a) m and (b) b ?

Consider the equation [latex] s={s}_{0}+{v}_{0}t+{a}_{0}{t}^{2}\text{/}2+{j}_{0}{t}^{3}\text{/}6+{S}_{0}{t}^{4}\text{/}24+c{t}^{5}\text{/}120, [/latex] where s is a length and t is a time. What are the dimensions and SI units of (a) [latex] {s}_{0}, [/latex] (b) [latex] {v}_{0}, [/latex] (c) [latex] {a}_{0}, [/latex] (d) [latex] {j}_{0}, [/latex] (e) [latex] {S}_{0}, [/latex] and (f) c ?

a. [latex] [{s}_{0}]=\text{L} [/latex] and units are meters (m); b. [latex] [{v}_{0}]={\text{LT}}^{-1} [/latex] and units are meters per second (m/s); c. [latex] [{a}_{0}]={\text{LT}}^{-2} [/latex] and units are meters per second squared (m/s 2 ); d. [latex] [{j}_{0}]={\text{LT}}^{-3} [/latex] and units are meters per second cubed (m/s 3 ); e. [latex] [{S}_{0}]={\text{LT}}^{-4} [/latex] and units are m/s 4 ; f. [latex] [c]={\text{LT}}^{-5} [/latex] and units are m/s 5 .

(a) A car speedometer has a 5% uncertainty. What is the range of possible speeds when it reads 90 km/h? (b) Convert this range to miles per hour. Note 1 km = 0.6214 mi.

A marathon runner completes a 42.188-km course in 2 h, 30 min, and 12 s. There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the percent uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?

a. 0.059%; b. 0.01%; c. 4.681 m/s; d. 0.07%, 0.003 m/s

The sides of a small rectangular box are measured to be 1.80 ± 0.1 cm, 2.05 ± 0.02 cm, and 3.1 ± 0.1 cm long. Calculate its volume and uncertainty in cubic centimeters.

When nonmetric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was used, where 1 lbm = 0.4539 kg. (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?

a. 0.02%; b. 1×10 4 lbm

The length and width of a rectangular room are measured to be 3.955 ± 0.005 m and 3.050 ± 0.005 m. Calculate the area of the room and its uncertainty in square meters.

A car engine moves a piston with a circular cross-section of 7.500 ± 0.002 cm in diameter a distance of 3.250 ± 0.001 cm to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.

a. 143.6 cm 3 ; b. 0.2 cm 3 or 0.14%

Challenge Problems

The first atomic bomb was detonated on July 16, 1945, at the Trinity test site about 200 mi south of Los Alamos. In 1947, the U.S. government declassified a film reel of the explosion. From this film reel, British physicist G. I. Taylor was able to determine the rate at which the radius of the fireball from the blast grew. Using dimensional analysis, he was then able to deduce the amount of energy released in the explosion, which was a closely guarded secret at the time. Because of this, Taylor did not publish his results until 1950. This problem challenges you to recreate this famous calculation. (a) Using keen physical insight developed from years of experience, Taylor decided the radius r of the fireball should depend only on time since the explosion, t , the density of the air, [latex] \rho , [/latex] and the energy of the initial explosion, E . Thus, he made the educated guess that [latex] r=k{E}^{a}{\rho }^{b}{t}^{c} [/latex] for some dimensionless constant k and some unknown exponents a , b , and c . Given that [E] = ML 2 T –2 , determine the values of the exponents necessary to make this equation dimensionally consistent. ( Hint : Notice the equation implies that [latex] k=r{E}^{\text{−}a}{\rho }^{\text{−}b}{t}^{\text{−}c} [/latex] and that [latex] [k]=1. [/latex]) (b) By analyzing data from high-energy conventional explosives, Taylor found the formula he derived seemed to be valid as long as the constant k had the value 1.03. From the film reel, he was able to determine many values of r and the corresponding values of t . For example, he found that after 25.0 ms, the fireball had a radius of 130.0 m. Use these values, along with an average air density of 1.25 kg/m 3 , to calculate the initial energy release of the Trinity detonation in joules (J). ( Hint : To get energy in joules, you need to make sure all the numbers you substitute in are expressed in terms of SI base units.) (c) The energy released in large explosions is often cited in units of “tons of TNT” (abbreviated “t TNT”), where 1 t TNT is about 4.2 GJ. Convert your answer to (b) into kilotons of TNT (that is, kt TNT). Compare your answer with the quick-and-dirty estimate of 10 kt TNT made by physicist Enrico Fermi shortly after witnessing the explosion from what was thought to be a safe distance. (Reportedly, Fermi made his estimate by dropping some shredded bits of paper right before the remnants of the shock wave hit him and looked to see how far they were carried by it.)

The purpose of this problem is to show the entire concept of dimensional consistency can be summarized by the old saying “You can’t add apples and oranges.” If you have studied power series expansions in a calculus course, you know the standard mathematical functions such as trigonometric functions, logarithms, and exponential functions can be expressed as infinite sums of the form [latex] \sum _{n=0}^{\infty }{a}_{n}{x}^{n}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+{a}_{3}{x}^{3}+\cdots , [/latex] where the [latex] {a}_{n} [/latex] are dimensionless constants for all [latex] n=0,1,2,\cdots [/latex] and x is the argument of the function. (If you have not studied power series in calculus yet, just trust us.) Use this fact to explain why the requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency. That is, it actually implies the arguments of standard mathematical functions must be dimensionless, so it is not really necessary to make this latter condition a separate requirement of the definition of dimensional consistency as we have done in this section.

Since each term in the power series involves the argument raised to a different power, the only way that every term in the power series can have the same dimension is if the argument is dimensionless. To see this explicitly, suppose [x] = L a M b T c . Then, [x n ] = [x] n = L an M bn T cn . If we want [x] = [x n ], then an = a, bn = b, and cn = c for all n. The only way this can happen is if a = b = c = 0.

OpenStax University Physics. Authored by : OpenStax CNX. Located at : https://cnx.org/contents/[email protected]:Gofkr9Oy@15 . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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27 4.6 Problem-Solving Strategies

Understand and apply a problem-solving procedure to solve problems using Newton’s laws of motion.

Problem-Solving Strategy for Newton’s Laws of Motion

Step 1. As usual, it is first necessary to identify the physical principles involved. Once it is determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation . Such a sketch is shown in Figure 1 (a). Then, as in Figure 1 (b), use arrows to represent all forces, label them carefully, and make their lengths and directions correspond to the forces they represent (whenever sufficient information exists).

Step 2. Identify what needs to be determined and what is known or can be inferred from the problem as stated. That is, make a list of knowns and unknowns. Then carefully determine the system of interest . This decision is a crucial step, since Newton’s second law involves only external forces. Once the system of interest has been identified, it becomes possible to determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 1 (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated earlier in this chapter, the system of interest depends on what question we need to answer. This choice becomes easier with practice, eventually developing into an almost unconscious process. Skill in clearly defining systems will be beneficial in later chapters as well.

Applying Newton’s Second Law

For example, if the system is accelerating in the horizontal direction, but it is not accelerating in the vertical direction, then you will have the following conclusions:

You will need this information in order to determine unknown forces acting in a system.

Section Summary

Draw a sketch of the problem.
Identify known and unknown quantities, and identify the system of interest. Draw a free-body diagram, which is a sketch showing all of the forces acting on an object. The object is represented by a dot, and the forces are represented by vectors extending in different directions from the dot. If vectors act in directions that are not horizontal or vertical, resolve the vectors into horizontal and vertical components and draw them on the free-body diagram.
Check your answer. Is the answer reasonable? Are the units correct?

Problems & Exercises

Using the free-body diagram:

Use Newton’s laws of motion
Use Newton’s laws since we are looking for forces.

This seems reasonable, since the applied tensions should be greater than the force applied to the tooth.

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A Logical Problem Solving Strategy

This page was developed by David DeMuth based on: Heller & Heller, "The Competent Problem Solver, A Strategy for Solving Problems in Physics", calculus version, 2nd ed., Minneapolis, MN: McGraw-Hill, 1995. If you are a faculty member or researcher and would like a complimentary copy of "The Competent Problem Solver", please contact McGraw-Hill Publishing @ 1-800-338-3987-3, or go to McGraw-Hill Higher Education Website .

Introduction

At one level, problem solving is just that, solving problems. Presented with a problem you try to solve it. If you have seen the problem before and you already know its solution, you can solve the problem by recall. Solving physics problems is not very different from solving any kind of problem. In your personal and professional life, however, you will encounter new and complex problems. The skillful problem solver is able to invent good solutions for these new problem situations. But how does the skillful problem solver create a solution to a new problem? And how do you learn to be a more skillful problem solver?

Research in the nature of problem solving has been done in a variety of disciplines such as physics, medical diagnosis, engineering, project design and computer programming. There are many similarities in the way experts in these disciplines solve problems. The most important result is that experts follow a general strategy for solving all complex problems. If you practice and learn this general strategy you will be successful in this course. In addition, you will become familiar with a general strategy fro solving problems that will be useful in your chosen profession.

A Logical Problem-Solving Strategy

Experts solve real problems in several steps. Getting started is the most difficult step. In the first and most important step, you must accurately visualize the situation, identify the actual problem , and comprehend the problem . At first you must deal with both the qualitative and quantitative aspects of the problem. You must interpret the problem in light of your own knowledge and experience; ie. Understanding . This enables you to decide what information is important, what information can be ignored, and what additional information may be needed, even though it was not explicitly provided. In this step it is also important to draw a picture of the problem situation. A picture is worth a thousand words if, of course, it is the right picture. (If a picture is worth a thousand words, and words are a dime a dozen, then what is a pictures monetary value?) In the second step, you must represent the problem in terms of formal concepts and principles, whether these are concepts of architectural design, concepts of medicine, or concepts of physics. These formal concepts and principles enable you to simplify a complex problem to its essential parts, making the search for a solution easier. Third, you must use your representation of the problem to plan a solution . Planning results in an outline of the logical steps required to obtain a solution. In many cases the logical steps are conveniently expressed as mathematics. Forth, you must determine a solution by actually executing the logical steps outlined in your plan. Finally, you must evaluate how well the solution resolves the original problem.

The general strategy can be summarized in terms of five steps:

The strategy begins with the qualitative aspects of a problem and progresses toward the quantitative aspects of a problem. Each step uses information gathered in the previous step to translate the problem into more quantitative terms. These steps should make sense to you. You have probably used a similar strategy when you have solved problems before.

A Physics-Specific Strategy

Each profession has its own specialized knowledge and patterns of thought. The knowledge and thought processes that you use in each of the steps will depend on the discipline in which you operate. Taking into account the specific nature of physics, we choose to label and interpret the five steps of the general problem solving strategy as follows:

Focus the Problem:

Describe the physics:, plan the solution:, execute the plan:, evaluate the answer:.

Consider each step as a translation of the previous step into a slightly different language. You begin with the full complexity of real objects interacting in the real world and through a series of steps arrive at a simple and precise mathematical expression.

The five-step strategy represents an effective way to organize your thinking to produce a solution based on your best understanding of physics. The quality of the solution depends on the knowledge that you use in obtaining the solution. Your use of the strategy also makes it easier to look back through your solution to check for incorrect knowledge and assumptions. That makes it an important tool for learning physics. If you learn to use the strategy effectively, you will find it a valuable tool to use for solving new and complex problems. After all, those are the ones that you will be hired to solve in your chosen profession.

Open access
Published: 03 January 2020

Students’ problem-solving strategies in qualitative physics questions in a simulation-based formative assessment

Mihwa Park ORCID: orcid.org/0000-0002-9549-9515 1

Disciplinary and Interdisciplinary Science Education Research volume 2 , Article number: 1 ( 2020 ) Cite this article

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Previous studies on quantitative physics problem solving have been concerned with students’ using equations simply as a numerical computational tool. The current study started from a research question: “How do students solve conceptual physics questions in simulation-based formative assessments?” In the study, three first-year college students’ interview data were analyzed to characterize their problem-solving strategies in qualitative physics questions. Prior to the interview, the participating students completed four formative assessment tasks in physics integrating computer simulations and questions. The formative assessment questions were either constructed-response or two-tiered questions related to the simulations. When interviewing students, they were given two or three questions from each task and asked to think aloud about the questions. The findings showed that students still used equations to answer the qualitative questions, but the ways of using equations differed between students. The study found that when students were able to connect variables to a physical process and to interpret relationships among variables in an equation, equations were used as explanatory or conceptual understanding tools, not just as computational tools.

Introduction

Since the new U.S. science standards, Next Generation Science Standards (NGSS), were released (NGSS Lead States, 2013 ), science assessments have been moving towards revealing students’ reasoning and their ability to apply core scientific ideas in solving problems (National Research Council, 2014 ; Pellegrino, 2013 ). Underwood, Posey, Herrington, Carmel, and Cooper ( 2018 ) suggested types of questions aligned with three-dimensional learning in A Framework for K-12 Science Education (National Research Council, 2012 ). These questions include constructed-response (CR) questions and two-tiered questions. Underwood et al. ( 2018 ) also argued that questions should address core and cross-cutting ideas and ask students to consider how scientific phenomena occur so that they can construct explanations and engage in argumentation. The underlying assumption of this approach could be that qualitative explanation questions (i.e., questions that ask students to explain qualitatively) reveal students’ reasoning and understanding of core scientific concepts better than do traditional multiple-choice and simple-calculation questions. Numerous studies in physics education have examined students’ problem-solving strategies, including studies that have identified differences in the problem-solving strategies employed by experts and novices. Experts tend to start by using general scientific principles to analyze problems conceptually, while novices tend to start by selecting equations and plugging in numbers (Larkin, McDermott, Simon, & Simon, 1980 ; Maloney, 1994 ; Simon & Simon, 1978 ). Thus, giving students opportunities to reason qualitatively about problems could help them to think like experts (van Heuvelen, 1991 ).

Another way to enhance students’ conceptual understanding of scientific ideas could be using computer simulations, because computer simulations help students visualize scientific phenomena that cannot be easily and accurately observed in real life. Many empirical studies support integrating computer simulations into assessments in order to promote students’ engagement in exploring scientific phenomena (de Jong & van Joolingen, 1998 ) and their conceptual understanding (Rutten, van Joolingen, & van der Veen, 2012 ; Trundle & Bell, 2010 ). For example, Quellmalz, Timms, Silberglitt, and Buckley ( 2012 ) developed a simulation-based science assessment, and found that the assessment was effective to reveal students’ knowledge and to find evidence of students’ reasoning. In the current study, computer simulations and conceptual qualitative questions were incorporated as integral parts of formative assessment to reveal students’ problem-solving strategies in answering qualitative physics questions. Therefore, the current study investigated students’ problem-solving strategies in physics, which offered them opportunities to elicit their reasoning by qualitatively explaining what would happen and why it would happen about a given physical situation.

Students’ strategies to solving physics problems

Early research on physics problem solving identified differences between experts and novices in their problem-solving strategies. For example, experts’ knowledge is organized into structures; thus, they demonstrate the effective use of sophisticated strategies to solve problems (Gick, 1986 ). Conversely, novices tend to describe physics problems at best in terms of equations, and spontaneously use superficial analogies (Gick, 1986 ). Experts also effectively use the problem decomposition strategy: breaking down a problem into subproblems, then solving each subproblem and combining them to form the final solution (Dhillon, 1998 ). They also apply relevant principle and laws to solve problems (Chi, Feltovich, & Glaser, 1981 ; Dhillon, 1998 ). By contrast, novices start with selecting equations and cue into surface features (Chi et al., 1981 ). A common finding from studies on differences between experts and novices in problem solving (e.g., Chi et al., 1981 ; Dhillon, 1998 ; Gick, 1986 ; Larkin et al., 1980 ) is that experts demonstrate their expertise in conceptual analysis of the problems using scientific principles and laws, then translate the problem into relevant mathematical equations, while novices jump to mathematical manipulations without the prior process of conceptual analysis (Larkin et al., 1980 ).

Huffman ( 1997 ) incorporated the results of studies on the differences in problem solving between experts and novices to formulate explicit problem-solving procedures for students. The procedures include five steps: (a) performing a qualitative analysis of the problem situation; (b) translating the conceptual analysis into a simplified physics description; (c) translating the physics description into specific mathematical equations to plan the solution; (d) combining the equations according to the plan; and (e) evaluating the solution to ensure it is reasonable and complete (Huffman, 1997 ). In essence, the procedure is designed to ensure students will conceptually reason about the problem first, using relevant scientific principles and laws, before jumping to selecting mathematical equations.

It is possible that students’ problem-solving strategies are influenced by problem representations (verbal, mathematical, graphical, etc.). Kohl and Finkelstein ( 2006 ) investigated how problem representations and student performance were related, and found that student strategies to solve physics problems often varied with different representations. They also found that not only problem representations but a number of other things, including prior knowledge and experience in solving problems from their previous classes, also influenced students’ performance, especially in the case of low-performing students. When asking students not to calculate a science question but to explain it conceptually, a study found that they still used equations or numerical values to solve the problems, indicating that they translated a conceptual qualitative question into a quantitative one (De Cock, 2012 ). Although students may succeed in calculating values in physics problems, it doesn’t always mean that they have good conceptual understanding of the questions (McDermott, 1991 ).

While earlier studies have been concerned with students’ using equations without conceptual understanding when solving problems, mathematical modeling plays a critical role in the epistemology in physics (Redish, 2017 ). Redish emphasized the importance of connecting physical meaning to mathematical representation when solving problems, because in physics, mathematical equations are linked to physical systems, and an equation contains packed conceptual knowledge. Thus, in physics, equations are not only computational tools but also symbolic representations of logical reasoning (Redish, 2005 , 2017 ). As such, students are expected to incorporate mathematical equations into their intuition of the physical world to conceptualize the physical system (Redish & Smith, 2008 ). In a study of students’ quantitative problem solving, Kuo, Hull, Gupta, and Elby ( 2012 ) pointed out the importance of connecting mathematical symbols to conceptual reasoning. Their study was conducted based on an assumption that equations should be blended with conceptual meaning in physics, which turned the attention of researchers on problem solving from how students select equations to how they use the equations. Kuo et al. ( 2012 ) concluded that blending of mathematical operations with conceptual reasoning constitutes good problem solving; thus, this blended process should be a part of problem-solving expertise in physics.

Using computer simulations as an assessment tool

Given that visualization plays a central role in the conceptualization process of physics (Kozhevnikov, Motes, & Hegarty, 2007 ), previous studies have used computer simulations to visualize scientific phenomena, especially those that cannot be accurately observed in real life, and reported their positive effect on students’ learning outcomes (Ardac & Akaygun, 2004 ; Dori & Hameiri, 2003 ). Using computer simulation to facilitate student learning in science was found to be especially effective on student performance, motivation (Rutten et al., 2012 ), and conceptual change (Smetana & Bell, 2012 ).

Computer simulation can be used not only as an instructional tool but also as an assessment tool. For example, Park, Liu, and Waight ( 2017 ) developed computer simulations for U.S. high school chemistry classes to help students conceptualize scientific phenomena, and then integrated the simulations into formative assessments with questions related to the simulations. Quellmalz et al. ( 2012 ) and Srisawasdi and Panjaburee ( 2015 ) also embedded computer simulations into formative assessments for use in science classrooms, and demonstrated positive effects on students’ performance compared to students who experienced only traditional assessments (e.g., paper-and-pencil tests). While many empirical studies have been done to investigate problem-solving strategies of students, there is a lack in studies on students’ strategies to solve physics problems when computer simulations were used as a visual representation and conceptual explanation questions were asked to reveal the students’ reasoning. This study addresses the gap in the body of literature by investigating students’ strategies in solving conceptual explanation questions in a simulation-based formative assessment.

Research procedure and participants

In the study, computer simulations and formative assessment questions were integrated into a web-based formative assessment system for online administration, which allowed students to use it at their convenience (Park, 2019 ). The formative assessment questions were either CR or two-tiered questions related to the simulations. A two-tiered question consists of a simple multiple-choice (MC) question and a justification question for which students write a justification for their answer to the MC question. This format of question was suggested to diagnose possible misconceptions held by students (Treagust, 1985 ) and to provide information about students’ reasoning behind their selected responses (Gurel, Eryılmaz, & McDermott, 2015 ). Computer simulations were selected from the Physics Education Technology (PhET) project ( https://phet.colorado.edu/ ) and embedded into the formative assessment system. The assessments targeted students’ conceptual understanding in physics, thus they were not asked to calculate any values or to demonstrate their mathematical competence (Park, 2019 ). Specifically, the questions presented a scientific situation and asked students to predict what would happen; then the assessment system asked students to run a simulation, posing questions asking for explanation of the phenomena and comparison between their prior ideas and the observed phenomena. Figure 1 presents example questions and simulation for the energy conservation task. After answering the questions, students ran the simulation and responded to questions asking how the skater’s highest speed changed and why they think it happened using evidence found in the simulation.

Energy conservation task example questions

Initially, first-year college students were recruited from a calculus-based, introductory level physics course at a large, public university in the United States; no particular demographic was targeted during recruitment. The physics course was offered to students majoring in subjects related to science or engineering and covered mechanics, including kinematics and conservation of energy, so simulations were selected to align with the course content. After selecting simulations from the PhET project, related formative assessment questions were created. As previously mentioned, the questions first asked students to predict what would happen in a given situation. In this case, verbal (expressed in writing) and pictorial representations (including images, diagrams, or graphs) describing the situation were shown on the screen (Fig. 1 ). Next, after the students answered the questions, the simulations were enabled for the students to run, and they were asked to explain the results. In total, four formative assessment tasks were developed and implemented online, and each task contained from 14 to 17 questions. Topics for the four tasks were (1) motion in two dimensions, (2) the laws of motion, (3) motion in one dimension and friction, and (4) conservation of energy. Descriptions of the four tasks are presented below.

Task 1: Students explore what factors will affect an object’s projectile motion when firing a cannon.

Task 2: Students create an applied force such as pulling against or pushing an object and observe how it makes the object move.

Task 3: Students explore the forces at work when a person tries to push a filing cabinet on a frictionless or frictional surface.

Task 4: Students explore a skater’s motion on different shapes of tracks and explore the relationship between the kinetic energy and thermal energy of the skater.

After the participating students completed the online implementations of the four tasks, an interview invitation email was sent to the students who had completed all four tasks, did not skip any questions, and did not answer a question with an off-task response, but included responses that needed further clarification. Initially, we invited six students to clarify and elaborate on their responses so we could better understand what they were thinking. When scoring students’ written responses, some responses needed further clarification. For example, students mentioned that in projectile motion, “mass is not relative to time”; “the greater angle will create a larger x component of velocity in a projectile motion”; or “an object’s speed is broken up evenly resulting in more air time”. In case of the energy conservation task, the responses needing more clarification were “the speed did not change because speed does not depend on mass” or “because a skater’s total energy increases with increase in mass, her speed does not change”. Those responses were not clear to the author. Thus, the author decided to invite them to clarify their responses. During the interviews, the students’ verbal responses inspired the author to explore differences in their problem-solving strategies to answer conceptual physics questions. Three students especially, Alex, Christopher, and Blake (all pseudonyms), demonstrated noticeable differences in their problem-solving strategies; therefore, they are the focus of the analysis in the current study.

Interview context and protocols

Semi-structured interviews were conducted to investigate students’ reasoning when responding to conceptual physics questions. To this end, the students were given two or three questions from each task and asked to think aloud about the questions. After they verbally answered each question, they were given their original written responses to see if their answers had changed, and if so, to explain why. When students used mathematical equations or graphs in their explanations, they were asked to explain why they used those particular strategies and how the strategies helped them to answer the questions. Some example interview questions were; “Please read the question. Will you tell me your answer for the question?”, “How did you answer this question?”, “Could you clarify what this means?”, and “What did you mean by (specific terms that students used)?” Students were interviewed individually by two interviewers. The interviews, which took place in an interview room located at their university, each lasted an hour.

While the interviews were going on, the author wrote memos about the students’ strategies to answer the given questions and their misconceptions about science. Interviews were audio recorded and transcribed verbatim. The transcripts were initially analyzed to prepare and organize data into emergent themes. In this process, the memos were also used. As a result, three initial themes were developed: 1) students’ strategies to answer problems, 2) effects of the assessment on students’ learning, and 3) students’ misconceptions about science. In the study, the first theme—strategies to answer problems—was made a focus in the next level of analysis, as the students demonstrated noticeable differences in using equations to answer conceptual physics problems. After choosing the theme as a main focus, the author analyzed it by open coding the relevant parts of the transcripts of the individual student interviews (interviews about Tasks 1–3) to formulate possible characterizations of students’ problem-solving strategies, especially when they were using equations. The author constantly compared the characterizations to integrate and refine them (Strauss & Corbin, 1998 ). After that, the rest of each individual student interview transcript (interviews about Task 4) was analyzed, using the same categories to confirm the findings. Students’ drawings (i.e., graphs) used to explain their reasoning were also considered as a data source (Creswell, 2016 ). Once characterizations in students’ use of equations in qualitative physics question were identified and compared across cases, the analysis results were given to a physics education researcher to seek an external check (Creswell & Miller, 2000 ).

Previous studies on expert and novice problem-solving strategies were reflected in the design of the formative assessment questions. Specifically, it was hypothesized that conceptual explanation questions would help students think about the questions more conceptually, so that they would start to solve them using scientific concepts and laws. Therefore, short written questions in the tasks asked the students to explain or to justify their answers without using a formula. Nonetheless, when we were interviewing students, we found that they preferred to use equations and mathematical concepts when explaining physical situations. Although the three participating students commonly used equations or mathematical concepts in their explanations, how they used the equations or mathematical concepts differed. Detailed findings are presented below in three subsections representing patterns in problem-solving strategies. Formative assessment Tasks 1, 2, and 3 were designed to address the topic of Motion and Force, while Task 4 covered the topic of Energy Conservation. We analyzed interview data by these two topics. Note that two terms—formula and equation—were not differentiated in the analysis of data; instead, they were considered synonyms.

Alex’s case – using equations as a conceptual understanding tool

Motion and force.

When interviewing Alex, we asked him what would happen if a person pushed a box, then let it go (Task 2). He said, “If it is frictionless, the box will move forever with a constant velocity, and if friction exists, the speed will decrease and eventually the box will stop.” This answer was very similar to his original written response. Next, we asked what would happen to the box’s motion after another box was placed on top of it. Alex said, “I don’t know how to explain this without a formula.” Because the original questions had asked students not to use formulas, he assumed that he was not allowed to use one in this explanation, and obviously he was struggling to explain without it. We told him to use formulas whenever he wanted, and he quickly jumped into using one.

Alex: Resultant force equals mass times acceleration, so if you have a bigger mass. Uh, if the resultant force was 50N, that’s the force you applied, and then you had 10N in friction, for example, then the resultant force is 40. You had, if you had 20kg, the acceleration would be 2. If you had 50kg, the acceleration would be 4 over 5, which is 0.8, which is less than 2. So, the more mass you have the smaller the acceleration is going to be, as a result of the resultant force equals ma equation.

In this statement, Alex explained what would happen in the given situation with algebraic solutions, using F = ma equation, and concluded that mass would affect the object’s acceleration, as he demonstrated. He further described how the eq. ( F = ma ) helped him to explain the given physical situation.

Alex: If you use the formula, then it makes it much easier, because in real life, you never see something moving without friction, so it just clouds your judgment a bit.

In this statement, Alex described the role of equation for him as a conceptual understanding tool, especially in an ideal situation that is not observable in real life. This was something the author had not initially expected from the students during their interviews. When we asked Alex the next question in Task 3, his answer further supported the finding that equations helped him understand physical situations. Specifically, we asked, in a situation when a person was pushing a cabinet on either a frictionless or a frictional surface, what would happen to the cabinet’s motion and why.

Alex: The normal force is, the gravitational force cancels out the y , so the only thing acting on the—in the x -direction, which is the direction being pushed is the applied force, so as small of a force you apply to it, it’s still going to move it because there’s nothing opposing it…if there was friction, I agree that it won’t move. Because the friction, the friction is the coefficient of friction times the normal force, so, since it’s a really big object, it’s going to have a significant amount of friction acting on it.

In his verbal explanation, Alex used a mathematical concept and an equation to explain the given phenomenon, using the vector concept for two components of force and a mathematical equation for frictional force. Obviously, he found equations useful to make sense of physical situations and to explain his understanding to others. Notably, he started his answer by referring to the formula for kinetic friction force and used the formula as a tool to explain why the cabinet wouldn’t move on a frictional surface. His explanation again demonstrated that equations and mathematical concepts were useful to understanding and interpreting scientific phenomena, and not only as a simple computational tool, at least for Alex.

Conservation of energy

Task 4 was designed to investigate students’ conceptions of mechanical energy and its conservation. We asked Alex, when a skater is skateboarding on a track with no friction, what would happen to the skater’s highest speed as the skater’s mass increases? He again asked us if he could use equations. We confirmed that he was allowed to use equations anytime he wanted. Then he immediately started writing equations on the board (see Fig. 2 ). While he was writing, he explained each variable involved in the equations:

Alex: So, her initial, so, um, at the start, her initial energy is mgh + ½ mv 0 2 and then her final [writing on board] mgh + ½ mv f 2 , but the smaller thing to do is that they [mass] all cancel out, so the mass is really, it doesn’t play a role in the height or the velocity. And then, if you wanted to see how the conversion of energy works, if you were initially starting at the maximum height, whatever that is, you could do ½ mv 2 . At the start, her velocity is 0, at the top, so this cancels out, if we’re analyzing it at the bottom, which is her max speed, then this [ h ] is 0, and then you just do gh = v 2 . To find her velocity. Just looking at this, there’s no mass in this, so it doesn’t matter [the skater’s speed]. When you actually work it out, all the masses cancel out, so it doesn’t matter what the mass is, in reality, when you actually calculate it.

This response was different than his original written response to the same question: “If the skater has a larger mass, she will in turn have a larger gravitational potential energy since GPE [gravitational potential energy] has a direct relationship to mass. As a result and according to the principles of conservation of energy, the KE [kinetic energy] will be greater and thus the velocity will be greater.” In this original written response, Alex included a typical misconception that heavier objects fall faster (e.g., Gunstone, Champagne, & Klopfer, 1981 ; Lazonder & Ehrenhard, 2014 ); “If the skater has a larger mass…thus the velocity will be greater” (in his written response). This was the only case of a misconception found in Alex’s written responses. Notably, when he was using equations, he deduced that “it doesn’t matter what the mass is, in reality, when you actually calculate it” from his step-by-step problem-solving procedure using algebraic solutions. Although he solved the problem using equations through algebraic computation, he explained how the object’s velocity and height would change as the object moved: “At the start, her velocity is 0, at the top, so this cancels out, if we’re analyzing it at the bottom, which is her max speed, then this [ h ] is 0, and then you just do gh = v 2 .” Then he connected conceptual meaning to the equation: “Just looking at this, there’s no mass in this, so it doesn’t matter [the skater’s speed].” This confirmed that for Alex, equations were the first tool to make sense of physical situation. In other words, when he applied an equation to a physical situation, he considered variables related to specific situations, then connected conceptual meaning to the variables, which indicated that for him, equations played a role in analyzing and understanding physical situation.

Alex’s explanation

Christopher’s case – using equations as an explanatory tool

We asked Christopher a question—which tank shell would go farther when the initial angles for two tank shells were different (Task 1). In his original written response, he mentioned that “tank A (initial angle: 45 degree)’s speed is broken up more evenly and this results in more air time which leads to more distance covered in the x axis as well.” This answer was similar to Christopher’s thinking-aloud response, so we asked him to elaborate on what he meant by “speed is broken up more evenly.” Below is his response.

Christopher: Because the velocity is a vector quantity, the speed is still the same, but the velocity, the x and y axis are going to be more evenly split [for Tank A, with a 45-degree initial angle], whereas for Tank B [10-degree initial angle] it would have been almost all in the x axis and close to none in the y , so it wouldn’t get that much air time because the force of gravity still stays the same.

As seen in his response, Christopher deduced his answer from a mathematical concept (vector in this case) explaining why the 45-degree shell would have a greater horizontal range than the 10-degree shell one. His problem-solving strategy in the next questions (questions from Tasks 2 and 3) further confirmed that he used mathematical concepts and equations to explain physical situations. For example, when asked to compare two situations from Task 2—a person pushes a box and lets it go, and after placing another box on top of that, a person pushes both boxes and lets them go—Christopher immediately used F = ma and explained the situation.

Christopher: The velocity and the speed will be decreased because, when applying force, force is mass times acceleration. So, if it would be the same exact force with a higher mass, then the acceleration would have to go down significantly in order to keep the same number [force]. So, because of this, it wouldn’t speed up as much, so it would have a lower velocity after the force was applied [compared to the previous situation]. While you are pushing, the acceleration is constant. And if they let it go, there is no acceleration. Then speed will stay the same.

In his statement, he referred to F = ma , and explained why the box’s acceleration would be smaller when its mass increased using algebraic solutions, which is similar to Alex’s case. The difference is that Christopher’s explanation contained an interpretation of the relationship among velocity, acceleration, and applied force: “So, because of this, it wouldn’t speed up as much, so it would have a lower velocity after the force was applied.” This implies that Christopher did not just use the equation as a computational tool, but linked meanings to variables (force, velocity, mass, and acceleration) and interpreted a relationship among them. When we asked him a question from Task 3—when a person is pushing a cabinet, how will the cabinet’s velocity change after passing over the frictionless surface and traveling onto the surface with friction?—his answer reconfirmed that he considered the relationship among variables and gave conceptual meaning not only to the variables but also to the relationship, and used a mathematical concept as an important tool to interpret a physical situation.

Christopher: So, the velocity is 100% dependent on the acceleration, which depends on the force, and then in this scenario, it is the force at first, it has a much higher total net force in the x direction, whereas later on it decreases [on a frictional surface], but there’s still a positive net force in the x direction, so it will continue. The reason why it continues to speed up is because the acceleration is still positive. ‘Cause mass can’t really be negative so that [acceleration] is the only variable [to determine the change of velocity]. So, that’s why velocity continues to increase, it’s just not as much as before.

In his statement, Christopher did not interpret an individual variable separately; rather, he first considered the relationship between force, velocity, and acceleration using the concept of vector and scalar quantity (e.g., mass is not a vector quantity), and explained how each variable was influenced by the other variables’ changes. From the statements above, it is clear that Christopher reasoned through a physical process by interpreting relationships among variables and attaching conceptual meaning to the relationship and the variables.

When we asked Christopher about change in the skater’s highest speed when the skater’s mass increased, his original written and oral responses contained the common answer that the skater’s highest speed would stay the same because gravity acts on all objects equally: “the downward acceleration will be the same.” We further asked him about how total mechanical energy changes. His response is below.

Christopher: Her [the skater’s] mechanical energy would increase because the velocity would stay the same for kinetic, but the mass would go up, so it would make the answer higher. And it’s probably easier to think of it with GPE, can I use the formula to it?

Then he drew a formula on board (Fig. 3 ), and explained why the total mechanical energy would change.

Christopher: This is mg. Since these two [ gh, ½ ] stay the same for both cases, they can be canceled out. So then, these are the only variables in ME (mechanical energy), so if this [ m ] increases, then the whole system[’s energy] will increase, but it won’t change this [ v ] in the specific scenario. If you were to use the equations, once you were to set them equal to each other and solve for the final answer for each, they would still be the same, even though the mass is higher. But because it’s multiplied, you can cancel it [ m ] on both sides for that specific scenario, so it mainly just depends on the constant ½ and then the variable of height and the final velocity which would be the same for this case.

In his response, Christopher first explained the physical situation using the concept of energy and considered the situation as a system: “Her [the skater’s] mechanical energy would increase,” and “so if this [ m ] increases, then the whole system[‘s energy] will increase.” In order to prove why mass doesn’t affect the skater’s speed, he used an equation as an explanatory tool—“And it’s probably easier to think of it with GPE, can I use the formula to it?”—and showed that mass doesn’t affect the skater’s speed: “You can cancel it [ m ] on both sides for that specific scenario.” A noticeable difference from Alex’s approach is that Christopher used equations to prove his claim and to explain it in an easier way, while Alex used equations to make sense of the situation. In other words, equations were in play mainly as explanatory tools for Christopher, whereas they acted as conceptual understanding tools for Alex. Similarly to his previous responses to questions in Motion and Force, Christopher again demonstrated that he considered how all variables were related each other in the system, and attached meaning to the relationship and variables. Interestingly, he often used the phrase “specific scenario,” so we asked what it meant. Below is his response.

Christopher: The equations don’t really help because even though I see it and it’s in my head, but it’s not really useful if I don’t know the scenario. If it’s some problems, I know, are purposefully shaped to muddle it up, and make it purposefully confusing, but usually, when you run the scenario, in a program or in your head, it kind of takes out that confusing stuff.

The above response illustrated that Christopher conceptually interpreted the physical situation first, then translated equations into the physical situation. This strategy shared a commonality with Alex’s in that both students used equations in their explanations and connected how variables in the equations changed as the specific physical situation changed. At the same time, there was a difference between the two students. Christopher’s strategy started with an analysis of the situation, creating a physical scenario and then translating equations into the physical situation, while Alex mentioned relevant equations first, then connected them to the physical situation.

Christopher’s explanation. Note: ME = mechanical energy

Blake’s case – using equations as a computational tool

When we asked Blake which one would go farther when shot from a cannon, a tank shell or a baseball (when air resistance was negligible; Task 1), her original written response and her thinking-aloud response were similar: the mass of an object is not relative to its motion. When we asked her to explain why, she said:

Blake: Because I don’t see kg on the units at all [in the simulation]. kg is the unit for mass, kilograms, so, it’s not written as kg/m/s or something. You could easily compare it with units and mass is not part of the unit.

Her response was interesting in that she used the unit of velocity rather than acceleration. Also, she did not show her conceptual understanding of physical variables and their relationship as Christopher had done. We further asked her what factors should be changed to maximize the horizontal range of the projectile object, in order to elicit her reasoning about a projectile motion. Below is her response.

Blake: You need to throw it faster. Um, because, if you look at gun for example. It’s a really high velocity. So, you just see it going like straight because it’s just high velocity. And, um, if, if I’m throwing this phone, maximum distance it could go is like here [tosses phone, not very far]. Angle? I think…like the maximum distance for x axis and y axis is 45 degrees, but I think it should be a little lower. Around 45 but plus or minus 5 degrees, so like 40 degrees.

Interviewer: Why would you say that?

Blake: It doesn’t get that much time for vertical velocity, but the horizontal velocity will be faster.

In her response, Blake used real-life examples—shooting a gun and throwing a phone—as analogies to reason how to increase the horizontal range of a projectile object. However, when she threw the phone, she tossed it, which started it with a different initial angle from that of a bullet shot from a gun : “You just see it going like straight because it’s just high velocity.” Although she considered two directions of velocity when determining the optimal initial angle, she did not provide a scientifically reasonable explanation for why the initial angle should be lower than 45 degrees. It might be that Blake had learned that 45 degrees is the angle used to maximize range, but that she thought velocity would be more critical than the angle to determine the range, especially that the x -component of velocity would more important than the y -component because an object will fly faster horizontally than vertically when the x- component is greater. Thus, she lowered the initial angle a little bit. In the above statements, Blake did not demonstrate that she could consider the relationship between variables and link conceptual meanings to them (e.g., “Because I don’t see kg on the units at all” and “It doesn’t get that much time for vertical velocity, but the horizontal velocity will be faster”).

For the next question, we asked what would happen to the box’s motion after another box was placed on top of it. She said, “It would still be constant and stay at constant velocity in that motion.” We asked the question again, to clarify if she understood it.

Blake: Yeah. The velocity would be the same. After you let it go. So it will be at constant speed. And the force is proportional to the...wait, well acceleration is proportional to force and mass.

In her response, Blake attempted to apply Newton’s second law ( F = ma ), as the other two students had; however, she didn’t realize that acceleration is inversely proportional to mass, and therefore the velocity would be changed by the different acceleration. As a result, her response involved a misconception that mass doesn’t affect the speed of an object. In other words, she demonstrated her lack of understanding of the relationships between the variables (acceleration, velocity, mass, and force) involved in the situation. Her response to the questions confirmed that she explained scientific phenomena using variables in equations but failed to recognize the relationships among them. Instead she focused on individual variables, e.g., how acceleration will change as force changes, but did not explain how that would change velocity. She also did not explain how two components of velocity affect an object’s motion. Interestingly, she also used the unit of variable to justify her answer without applying conceptual meanings to it. For Blake, equations and units seemed to play important roles in explaining physical situations, but her connection of equations to physical situations was, at best, based on interpretations of individual variables.

When we asked Blake about change in the skater’s highest speed when the skater’s mass increased, her original written response was that her highest speed would increase because the mass of the skater would require more energy. When we interviewed her, her answer was different from her original response.

Blake: I think it should stay the same. I was thinking of the formula.

When we asked her to explain in more detail, she wrote an equation on the board (Fig. 4 ) and explained what it meant.

Blake: The highest point, because there won’t be any kinetic energy. And it’ll be mgh . Also ½ mv 2 and it [ m ] cancels out. It was exactly the same. The speed was the same. But—wasn’t there a bar graph [in the simulation]? Well, the total energy was bigger [in the simulation]. The total energy. But the total energy was same—no bigger.

Similarly to Alex, Blake used an equation to explain that the skater’s speed wouldn’t change because v doesn’t contain m after canceling out. However, she did not describe why kinetic energy is zero at the highest point and why potential energy is zero at the bottom. It might be that she just did not mention this, but it was obvious that she did not understand how the object’s mass affected the system: “But the total energy was same—no bigger.” We further asked her how the total mechanical energy of the skater would change when the skater’s mass increased. This time, she said, “Well, the total energy was bigger. ‘Cause energy depends on mass and either height or speed of a person.” As seen in the response, she thought of variables in equations of gravitational potential energy ( mgh ) and kinetic energy ( $ \frac{1}{2} $ mv 2 ). When asked why she previously had said the total mechanical energy would be the same, she answered, “because energy is always conserved.” This illustrated her misconception that the amount of energy should always be the same regardless of mass; however, when she considered variables in equations of PE and KE, she answered the question accurately. Throughout the interview, we found that Blake’s strategy to solve questions was consistent across different tasks; she used formulas and units as her first approach. However, a difference between Blake and the other two students is that although she used equations and variables, she did not explain how the variables influenced each other; and how they would change as a specific situation changed. In other words, she did not translate equations into physical situations nor link conceptual meanings to the variables and the relationships between them. The findings showed that for Blake, equations were more likely used as a simple computational tool.

Blake’s explanation

Disconnection of students’ problem-solving strategies from physics lecture

The three students all mentioned that they liked simulation-based questions. Alex said that the questions themselves made him think a lot, and running simulations also made him think more deeply: “Beforehand it [the task] just seems really simple, so you don’t put much thought into it. That’s easy. Just write it down, but then, once you run it, it makes you think about it more. So that’s cool too.”

As seen in his responses to the questions in the tasks, Alex used equations as conceptual understanding tools consistently across tasks. When we asked if he had learned this approach from his physics course, he said that his physics class heavily focused on solving problems but mostly by just reading off equations and plugging in numbers.

Alex: Physics is not about reading equations and stuff off a slide. It’s about working things by hand, and my professor, he has all the solutions to the problems in the book. He had them on a clear sheet of paper, and a Sharpie and then, so if he has problem 20, he puts problem 20 on the projector, and then he put that clean sheet there, and then he points to, oh here I did “v = a + blah blah,” so that’s really not effective at all in my opinion.

Christopher mentioned that the formative assessment would be very helpful for a lot of students, because it showed physical scenario. His physics class was more formula-based, with activities such as showing a formula and plugging in numbers to demonstrate how to solve a physics problem, which Christopher felt was disconnected from how he learned science. As he demonstrated, he learned best when he created a physical scenario, then translated it into equations. Alex also mentioned that physics is “about working things by hand,” which implies that he emphasized linking problem-solving procedures to physical situations. In Blake’s case, she mentioned that “It will help students to learn the concept better, but I think students will hate it [the formative assessment] because students will be like, ‘I don’t have time for this. It’s just that I am like too busy for this.’” In sum, the three students had a common opinion that the simulation-based formative assessment had helped them understand the given physical situation better, but the reasons why they liked it differed, as did their problem-solving strategies.

Discussion and conclusion

Previous studies on problem solving were concerned with students’ using equations simply as numerical computational tools by plugging in numbers. While experts tend to start with a conceptual analysis of problems using scientific principles and laws, novices start by selecting and manipulating equations without conceptual analysis (Larkin et al., 1980 ). The difference in solving problems might be more obvious in quantitative questions, in which a mathematically framed physics question may prompt students to use equations without conceptual understanding (Kohl & Finkelstein, 2006 ). The current study started from the research question “How do students solve conceptual physics questions in simulation-based formative assessments?” The findings showed that the students still used equations to answer the questions. However, their utilizations of equations were different. For example, Alex’s and Christopher’s strategies involved using equations to explain or interpret the given physical situation. To do so, they connected variables to physical situations and provided meanings to the variables and the relationships among the variables. Blake, however, used equations and units as tools to find answers for the questions without a clear connection of the variables and equations to the given physical situations. Christopher’s strategy was especially noticeable in that he used equations as effective explanatory tools for a physical situation. He started an analysis of the physical situation, then translated equations into the situation by creating a physical scenario in a system, such as how variables change as the situation changes, and how the variables are related to each other within the physical system. Alex’s explanations illustrated that he utilized equations to understand a physical situation. The difference between him and Christopher is that Alex used equations as major tools to analyze and understand the situation, while Christopher used them to effectively and easily explain the situation. Noticeably, Alex used algebraic computation processes using an equation to understand a given physical process.

Kuo et al. ( 2012 ) argued that linking conceptual reasoning to mathematical formalism indicates a more expert level of understanding and demonstrates robust solutions integrating conceptual and symbolic reasoning. They found that students used equations not just as computational tools but as tools to find conceptual shortcuts to solve physics problems. Although Kuo et al.’s study focused on quantitative problem solving, the current study revealed a similar finding where questions were created qualitatively without asking any calculations. Another difference from Kuo et al.’s study is that they provided an equation to students first, then asked them to explain the equation and apply the equation to a physical situation, whereas the current study provided a physical situation without any equations. As a conclusion, the current study supports that equations can be important in conceptualizing a physical situation by connecting conceptual meanings to equations. Therefore, mathematical equations can be used alternatively in problem solving (Kuo et al., 2012 ). Redish and Smith ( 2008 ) also illuminated the power of equations in solving physics problems and making sense of physical systems when students are able to link physical scenarios to mathematical equations. Thus, the connection of physical meaning to equations should be emphasized in teaching and learning physics in order to help students to conceptualize physical system (Redish & Smith, 2008 ).

Previous studies of quantitative physics problem solving have focused on using equations first when solving a physics question without a conceptual analysis of the problem situation, which indicated equations were in play as a simple computational tool. Although, the current study found a similar case, in which a student used equations as a simple computational tool, we also found that students used equations as a conceptual understanding or an effective explanatory tool. Indeed, using equations helped Alex realize his misconception and explain the situation accurately. While previous studies have emphasized performing a conceptual analysis first using scientific principles when solving a problem, this study argues the positive roles of using equations when it includes a connection between the equations and the physical situation. Therefore, this study contributes to the literature on physics problem solving in that equations can be used for students as tools for a conceptual understanding and as an explanatory tool. In this study, Christopher’s strategy was closer to the strategy used by experts, since he visualized a given situation to analyze by creating a physical scenario, then connected the relevant equations to the situation to explain the physical scenario. On the other hand, Alex used equations first to answer questions by connecting variables to the physical process through an algebraic solution process. Especially for Alex, equations facilitated his physical understanding of the problem and ability to explain the physical process. Although Alex and Blake used equations primarily as tools to answer questions, Blake did not demonstrate her interpretations of variables or the relationships among them in equations; nor did she connect variables to a physical situation. This indicated that her utilization of equations was closer to simple computational tools.

In conclusion, mathematical equations in physics were important when students were conceptually explaining a physical situation. It was revealed that using equations helped them explain a physical situation with more scientifically normative ideas. However, the ways they used equations differed between students. An equation could be an explanatory tool, a conceptual understanding tool, or a computational tool. The essence of the findings was that when students were able to connect variables to a physical process and to interpret relationships among variables in an equation, equations were in play as tools in understanding and explaining a physical situation. On the other hand, without interpretations of variables and connections to a physical situation, equations only served as simple computational tools. The study also found that students’ strategies to answer questions, especially conceptual ones, did not change with different topics in physics.

Implications and study limitations

As some students pointed out, their physics lectures demonstrated how to solve quantitative questions using equations as computational tools. As Christopher’s problem solving strategy was similar to the strategy used by experts, we suggest that his strategy be reflected in teaching physics. To be more specific, physics educators may provide an opportunity for students to visualize the physics phenomena. They could use models or computer simulations to help this procedure. Second, they should emphasize how equations are used to explain the phenomena as a “conceptual shortcut” (Kuo et al., 2012 , p. 39) by connecting equations and variables to the physical situation. In other words, as Alex and Christopher demonstrated, if physics instructions emphasize connections between physical meanings and mathematical expressions, it help students understand physical phenomenon. As we consider physics instructors as experts, perhaps in some cases their expert level of using equations was not reflected in their teaching. A future study topic would be to investigate the reason for the gap between physics experts’ strategies in solving physics problems and their teaching practices when demonstrating how to solve physics problems.

Although the findings of this study suggest an alternative way of using equations as an explanatory or a conceptual analysis tool for a physical situation, the findings might not be generalizable because the study context was limited to an introductory level physics course. Also, it is possible that topics for the tasks (kinematics and mechanical energy conservation) involving several equations might have influenced students’ strategies in answering questions. However, Redish ( 2017 ) emphasizes that a goal of physics is to create mathematical modeling (equations) that can predict and explain physical phenomena. Consequently, mathematical equations are included in physics topics and taught extensively in physics instruction especially in high school and college. We argue that students’ understanding of mathematical modeling in physics should not be considered as a following step after conceptual understanding of scientific principles. Instead, we support the claim that blending of physical meaning with mathematical operations should be emphasized in teaching physics (Kuo et al., 2012 ; Redish, 2005 , 2017 ). We also suggest that future studies should investigate how students’ strategies to answer questions are different in other topics, such as thermodynamics or electricity and magnetism.

Availability of data and materials

Interview data are not available for public.

Abbreviations

Constructed-Response

Gravitational Potential Energy

Kinetic Energy

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1.8: Solving Problems in Physics

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Learning Objectives

Describe the process for developing a problem-solving strategy.
Explain how to find the numerical solution to a problem.
Summarize the process for assessing the significance of the numerical solution to a problem.

Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You often need to decide which direction is positive and note that on your sketch. When you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.
Make a list of what is given or can be inferred from the problem as stated (identify the “knowns”) . Many problems are stated very succinctly and require some inspection to determine what is known. Drawing a sketch be very useful at this point as well. Formally identifying the knowns is of particular importance in applying physics to real-world situations. For example, the word stopped means the velocity is zero at that instant. Also, we can often take initial time and position as zero by the appropriate choice of coordinate system.
Identify exactly what needs to be determined in the problem (identify the unknowns). In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help identify the unknowns.
Determine which physical principles can help you solve the problem . Since physical principles tend to be expressed in the form of mathematical equations, a list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all the other variables are known—so you can solve for the unknown easily. If the equation contains more than one unknown, then additional equations are needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.

Significance

Check your units . If the units of the answer are incorrect, then an error has been made and you should go back over your previous steps to find it. One way to find the mistake is to check all the equations you derived for dimensional consistency. However, be warned that correct units do not guarantee the numerical part of the answer is also correct.
Check the answer to see whether it is reasonable. Does it make sense? This step is extremely important: –the goal of physics is to describe nature accurately. To determine whether the answer is reasonable, check both its magnitude and its sign, in addition to its units. The magnitude should be consistent with a rough estimate of what it should be. It should also compare reasonably with magnitudes of other quantities of the same type. The sign usually tells you about direction and should be consistent with your prior expectations. Your judgment will improve as you solve more physics problems, and it will become possible for you to make finer judgments regarding whether nature is described adequately by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to solve a problem mechanically.
Check to see whether the answer tells you something interesting. What does it mean? This is the flip side of the question: Does it make sense? Ultimately, physics is about understanding nature, and we solve physics problems to learn a little something about how nature operates. Therefore, assuming the answer does make sense, you should always take a moment to see if it tells you something about the world that you find interesting. Even if the answer to this particular problem is not very interesting to you, what about the method you used to solve it? Could the method be adapted to answer a question that you do find interesting? In many ways, it is in answering questions such as these science that progresses.

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Modified crayfish optimization algorithm for solving multiple engineering application problems

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Published: 24 April 2024
Volume 57 , article number 127 , ( 2024 )

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Heming Jia 1 ,
Xuelian Zhou 1 ,
Jinrui Zhang 1 ,
Laith Abualigah 2 , 3 ,
Ali Riza Yildiz 4 &
Abdelazim G. Hussien 5

Crayfish Optimization Algorithm (COA) is innovative and easy to implement, but the crayfish search efficiency decreases in the later stage of the algorithm, and the algorithm is easy to fall into local optimum. To solve these problems, this paper proposes an modified crayfish optimization algorithm (MCOA). Based on the survival habits of crayfish, MCOA proposes an environmental renewal mechanism that uses water quality factors to guide crayfish to seek a better environment. In addition, integrating a learning strategy based on ghost antagonism into MCOA enhances its ability to evade local optimality. To evaluate the performance of MCOA, tests were performed using the IEEE CEC2020 benchmark function and experiments were conducted using four constraint engineering problems and feature selection problems. For constrained engineering problems, MCOA is improved by 11.16%, 1.46%, 0.08% and 0.24%, respectively, compared with COA. For feature selection problems, the average fitness value and accuracy are improved by 55.23% and 10.85%, respectively. MCOA shows better optimization performance in solving complex spatial and practical application problems. The combination of the environment updating mechanism and the learning strategy based on ghost antagonism significantly improves the performance of MCOA. This discovery has important implications for the development of the field of optimization.

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Avoid common mistakes on your manuscript.

1 Introduction

For a considerable period, engineering application problems have been widely discussed by people. At present, improving the modern scientific level of engineering construction has become the goal of human continuous struggle, including constrained engineering design problems (Zhang et al. 2022a ; Mortazavi 2019 ) affected by a series of external factors and feature selection problems (Kira and Rendell 1992 ), and so on. Constrained engineering design problems refers to the problem of achieving optimization objectives and reducing calculation costs under many external constraints, which is widely used in mechanical engineering (Abualigah et al. 2022 ), electrical engineering (Razmjooy et al. 2021 ), civil engineering (Kaveh 2017 ), chemical engineering (Talatahari et al. 2021 ) and other engineering fields, such as workshop scheduling (Meloni et al. 2004 ), wind power generation (Lu et al. 2021 ), and UAV path planning (Belge et al. 2022 ), parameter extraction of photovoltaic models(Zhang et al. 2022b ; Zhao et al. 2022 ), Optimization of seismic foundation isolation system (Kandemir and Mortazavi 2022 ), optimal design of RC support foundation system of industrial buildings (Kamal et al. 2023 ), synchronous optimization of fuel type and external wall insulation performance of intelligent residential buildings (Moloodpoor and Mortazavi 2022 ), economic optimization of double-tube heaters (Moloodpoor et al. 2021 ).

Feature selection is the process of choosing specific subsets of features from a larger set based on defined criteria. In this approach, each original feature within the subset is individually evaluated using an assessment function. The aim is to select pertinent features that carry distinctive characteristics. This selection process reduces the dimensionality of the feature space, enhancing the model's generalization ability and accuracy. The ultimate goal is to create the best possible combination of features for the model. By employing feature selection, the influence of irrelevant factors is minimized. This reduction in irrelevant features not only streamlines the computational complexity but also reduces the time costs associated with processing the data. Through this method, redundant and irrelevant features are systematically removed from the model. This refinement improves the model’s accuracy and results in a higher degree of fit, ensuring that the model aligns more closely with the underlying data patterns.

In practical applications of feature selections, models are primarily refined using two main methods: the filter (Cherrington et al. 2019 ) and wrapper (Jović et al. 2015 ) techniques. The filter method employs a scoring mechanism to assess and rank the model's features. It selects the subset of features with the highest scores, considering it as the optimal feature combination. On the other hand, the wrapper method integrates the selection process directly into the learning algorithm. It embeds the feature subset evaluation within the learning process, assessing the correlation between the chosen features and the model. In recent years, applications inspired by heuristic algorithms can be seen everywhere in our lives and are closely related to the rapid development of today's society. These algorithms play an indispensable role in solving a myriad of complex engineering problems and feature selection challenges. They have proven particularly effective in addressing spatial, dynamic, and random problems, showcasing significant practical impact and tangible outcomes.

With the rapid development of society and science and technology, through continuous exploitation and exploration in the field of science, more and more complex and difficult to describe multi-dimensional engineering problems also appear in our research process. Navigating these complexities demands profound contemplation and exploration. While traditional heuristic algorithms have proven effective in simpler, foundational problems, they fall short when addressing the novel and intricate multi-dimensional challenges posed by our current scientific landscape and societal needs. Thus, researchers have embarked on a journey of continuous contemplation and experimentation. By cross-combining and validating existing heuristic algorithms, they have ingeniously devised a groundbreaking solution: Metaheuristic Algorithms (MAs) (Yang 2011 ). This innovative approach aims to tackle the complexities of our evolving problems, ensuring alignment with the rapid pace of social and technological development. MAs is a heuristic function based algorithm. It works by evaluating the current state of the problem and possible solutions to guide the algorithm in making choices in the search space. MAs improves the efficiency and accuracy of the problem solving process by combining multiple heuristic functions and updating the search direction at each step based on their weights. The diversity of MAs makes it a universal problem solver, adapting to the unique challenges presented by different problem domains. Essentially represents a powerful paradigm shift in computational problem solving, providing a powerful approach to address the complexity of modern engineering and scientific challenges. Compared with traditional algorithms, MAs has made great progress in finding optimal solutions, jumping out of local optima, and overcoming convergence difficulties in the later stage of solution through the synergy of different algorithms. These enhancements mark a significant progress, which not only demonstrates the adaptability of the scientific method, but also emphasizes the importance of continuous research and cooperation. It also has the potential to radically solve problems in domains of complex engineering challenges, enabling researchers to navigate complex problem landscapes with greater accuracy and efficiency.

Research shows that MAs are broadly classified into four different research directions: swarm-based, natural evolution-based, human-based, and physics-based. These categories include a wide range of innovative problem-solving approaches, each drawing inspiration from a different aspect of nature, human behavior, or physical principles. Researchers exploration these different pathways to solve complex challenges and optimize the solutions efficiently. First of all, the swarm-based optimization algorithm is the optimization algorithm that uses the wisdom of population survival to solve the problem. For example, Particle Swarm Optimization Algorithm (PSO) (Wang et al. 2018a ) is an optimization algorithm based on the group behavior of birds. PSO has a fast search speed and is only used for real-valued processing. However, it is not good at handling discrete optimization problems and has fallen into local optimization. Artificial Bee Colony Optimization Algorithm (ABC) (Jacob and Darney 2021 ) realizes the sharing and communication of information among individuals when bees collect honey according to their respective division of labor. In the Salp Swarm Algorithm (SSA) (Mirjalili et al. 2017 ), individual sea squirts are connected end to end and move and prey in a chain, and follow the leader with followers according to a strict “hierarchical” system. Ant Colony Optimization Algorithm (ACO) (Dorigo et al. 2006 ), ant foraging relies on the accumulation of pheromone on the path, and spontaneously finds the optimal path in an organized manner.

Secondly, a natural evolutionary algorithm inspired by the law of group survival of the fittest, an optimization algorithm that finds the best solution by preserving the characteristics of easy survival and strong individuals, such as: Genetic Programming Algorithm (GP) (Espejo et al. 2009 ), because biological survival and reproduction have certain natural laws, according to the structure of the tree to deduce certain laws of biological genetic and evolutionary process. Evolutionary Strategy Algorithm (ES) (Beyer and Schwefel 2002 ), the ability of a species to evolve itself to adapt to the environment, and produce similar but different offspring after mutation and recombination from the parent. Differential Evolution (DE) (Storn and Price 1997 ) eliminates the poor individuals and retains the good ones in the process of evolution, so that the good ones are constantly approaching the optimal solution. It has a strong global search ability in the initial iteration, but when there are fewer individuals in the population, individuals are difficult to update, and it is easy to fall into the local optimal. The Biogeography-based Optimization Algorithm (BBO) (Simon 2008 ), influenced by biogeography, filters out the global optimal value through the iteration of the migration and mutation of species information.

Then, Human-based optimization algorithms are optimization algorithms that take advantage of the diverse and complex human social relationships and activities in a specific environment to solve problems, such as: The teaching–learning-based Optimization (TLBO) (Rao and Rao 2016 ) obtained the optimal solution by simulating the Teaching relationship between students and teachers. It simplifies the information sharing mechanism within each round, and all evolved individuals can converge to the global optimal solution faster, but the algorithm often loses its advantage when solving some optimization problems far from the origin. Coronavirus Mask Protection Algorithm (CMPA) (Yuan et al. 2023 ), which is mainly inspired by the self-protection process of human against coronavirus, establishes a mathematical model of self-protection behavior and solves the optimization problem. Cultural Evolution Algorithm (CEA) (Kuo and Lin 2013 ), using the cultural model of system thinking framework for exploitation to achieve the purpose of cultural transformation, get the optimal solution. Volleyball Premier League Algorithm (VPL) (Moghdani and Salimifard 2018 ) simulates the process of training, competition and interaction of each team in the volleyball game to solve the global optimization problem.

Finally, Physics-based optimization algorithm is an optimization algorithm that uses the basic principles of physics to simulate the physical characteristics of particles in space to solve problems. For example, Snow Ablation Algorithm (SAO) (Deng and Liu 2023 ), inspired by the physical reaction of snow in nature, realizes the transformation among snow, water and steam by simulating the sublation and ablation of snow. RIME Algorithm (RIME) (Su et al. 2023 ) is a exploration and exploitation of mathematical model balance algorithm based on the growth process of soft rime and hard rime in nature. Central Force Optimization Algorithm (CFO) (Formato 2007 ), aiming at the problem of complex calculation of the initial detector, a mathematical model of uniform design is proposed to reduce the calculation time. Sine and cosine algorithm (SCA) (Mirjalili 2016 ) establishes mathematical models and seeks optimal solutions based on the volatility and periodicity characteristics of sine and cosine functions. Compared with the candidate solution set of a certain scale, the algorithm has a strong search ability and the ability to jump out of the local optimal, but the results of some test functions fluctuate around the optimal solution, and there is a certain precocious situation, and the convergence needs to be improved.

While the original algorithm is proposed, many improved MAs algorithms are also proposed to further improve the optimization performance of the algorithm in practical application problems, such as: Yujun-Zhang et al. combined the arithmetic optimization algorithm (AOA) with the Aquila Optimizer(AO) algorithm to propose a new meta-heuristic algorithm (AOAAO) (Zhang et al. 2022c ). CSCAHHO algorithm (Zhang et al. 2022d ) is a new algorithm obtained by chaotic mixing of sine and cosine algorithm (SCA) and Harris Hqwk optimization algorithm (HHO). Based on LMRAOA algorithm proposed to solve numerical and engineering problems (Zhang et al. 2022e ). Yunpeng Ma et al. proposed an improved teaching-based optimization algorithm to artificially reduce NOx emission concentration in circulating fluidized bed boilers (Ma et al. 2021 ). The improved algorithm SOS(MSOS) (Kumar et al. 2019 ), based on the natural Symbiotic search (SOS) algorithm, improves the search efficiency of the algorithm by introducing adaptive return factors and modified parasitic vectors. Modified beluga whale optimization with multi-strategies for solving engineering problems (MBWO) (Jia et al. 2023a ) by gathering Beluga populations for feeding and finding new habitats during long-distance migration. Betul Sultan Yh-ld-z et al. proposed a novel hybrid optimizer named AO-NM, which aims to optimize engineering design and manufacturing problems (Yıldız et al. 2023 ).

The Crayfish Optimization Algorithm (COA) (Jia et al. 2023b ) is a novel metaheuristic algorithm rooted in the concept of population survival wisdom, introduced by Heming Jia et al. in 2023. Drawing inspiration from crayfish behavior, including heat avoidance, competition for caves, and foraging, COA employs a dual-stage strategy. During the exploration stage, it replicates crayfish searching for caves in space for shelter, while the exploitation stage mimics their competition for caves and search for food. Crayfish, naturally averse to dry heat, thrive in freshwater habitats. To simulate their behavior and address challenges related to high temperatures and food scarcity, COA incorporates temperature variations into its simulation. By replicating crayfish habits, the algorithm dynamically adapts to environmental factors, ensuring robust problem-solving capabilities. Based on temperature fluctuations, crayfish autonomously select activities such as seeking shelter, competing for caves, and foraging. When the temperature exceeds 30°C, crayfish instinctively seek refuge in cool, damp caves to escape the heat. If another crayfish is already present in the cave, a competition ensues for occupancy. Conversely, when the temperature drops below 30°C, crayfish enter the foraging stage. During this phase, they make decisions about food consumption based on the size of the available food items. COA achieves algorithmic transformation between exploration and exploitation stages by leveraging temperature variations, aiming to balance the exploration and exploitation capabilities of the algorithm. However, COA solely emulates the impact of temperature on crayfish behavior, overlooking other significant crayfish habits, leading to inherent limitations. In the latter stages of global search, crayfish might cluster around local optimum positions, restricting movement. This hampers the crayfish's search behavior, slowing down convergence speed, and increasing the risk of falling into local optima, thereby making it challenging to find the optimal solution.

In response to the aforementioned challenges, this paper proposes a Modified Crayfish Optimization Algorithm (MCOA). MCOA introduces an environmental update mechanism inspired by crayfish's preference for living in fresh flowing water. MCOA incorporates crayfish's innate perception abilities to assess the quality of the surrounding aquatic environment, determining whether the current habitat is suitable for survival. The simulation of crayfish crawling upstream to find a more suitable aquatic environment is achieved by utilizing adaptive flow factors and leveraging the crayfish's second, third foot perceptions to determine the direction of water flow.This method partially replicates the survival and reproduction behavior of crayfish, ensuring the continual movement of the population. It heightens the randomness within the group, widens the search scope for crayfish, enhances the algorithm's exploration efficiency, and effectively strengthens the algorithm’s global optimization capabilities. Additionally, the ghost opposition-based learning strategy (Jia et al. 2023c ) is implemented to introduce random population initialization when the algorithm becomes trapped in local optima. This enhancement significantly improves the algorithm's capability to escape local optima, promoting better exploration of the solution space. After the careful integration of the aforementioned two strategies, the search efficiency and predation speed of the crayfish algorithm experience a substantial improvement. Moreover, the algorithm's convergence rate and global optimization ability are significantly enhanced, leading to more effective and efficient problem-solving capabilities.

In the experimental section, we conducted a comprehensive comparison between MCOA and nine other metaheuristic algorithms. We utilized the IEEE CEC2020 benchmark function to evaluate the performance of the algorithm. The evaluation involved statistical methods such as the Wilcoxon rank sum test and Friedman test to rank the averages, validating the efficiency of the MCOA algorithm and the effectiveness of the proposed improvements. Furthermore, MCOA was applied to address four constrained engineering design problems as well as the high-dimensional feature selection problem using the wrapper method. These practical applications demonstrated the practicality and effectiveness of MCOA in solving real-world engineering problems.

The main contributions of this paper are as follows:

In the environmental renewal mechanism, the water quality factor and roulette wheel selection method are introduced to simulate the process of crayfish searching for a more suitable water environment for survival.

The introduction of the ghost opposition-based learning strategy enhances the randomness of crayfish update locations, effectively preventing the algorithm from getting trapped in local optima, and improving the overall global optimization performance of the algorithm.

The fixed value of food intake is adaptively adjusted based on the number of evaluations, enhancing the algorithm's capacity to escape local optima. This adaptive change ensures a more dynamic exploration of the solution space, improving the algorithm's overall optimization effectiveness.

The MCOA’s performance is compared with nine metaheuristics, including COA, using the IEEE CEC2020 benchmark function. The comparison employs the Wilcoxon rank sum test and Friedman test to rank the averages, providing evidence for the efficiency of MCOA and the effectiveness of the proposed improvements.

The application of MCOA to address four constrained engineering design problems and the high-dimensional feature selection problem using the wrapper method demonstrates the practicality and effectiveness of MCOA in real-world applications.

The main structure of this paper is as follows, the first part of the paper serves as a brief introduction to the entire document, providing an overview of the topics and themes that will be covered. In the second part, the paper provides a comprehensive summary of the Crayfish Optimization Algorithm (COA). In the third part, a modified crawfish optimization algorithm (MCOA) is proposed. By adding environment updating mechanism and ghost opposition-based learning strategy, MCOA can enhance the global search ability and convergence speed to some extent. Section four shows the experimental results and analysis of MCOA in IEEE CEC2020 benchmark functions. The fifth part applies MCOA to four kinds of constrained engineering design problems. In Section six, MCOA is applied to the high-dimensional feature selection problem of wrapper methods to demonstrate the effectiveness of MCOA in practical application problems. Finally, Section seven concludes the paper.

2 Crayfish optimization algorithm (COA)

Crayfish is a kind of crustaceans living in fresh water, its scientific name is crayfish, also called red crayfish or freshwater crayfish, because of its food, fast growth rate, rapid migration, strong adaptability and the formation of absolute advantages in the ecological environment. Changes in temperature often cause changes in crayfish behavior. When the temperature is too high, crayfish choose to enter the cave to avoid the damage of high temperature, and when the temperature is suitable, they will choose to climb out of the cave to forage. According to the living habits of crayfish, it is proposed that the three stages of summer, competition for caves and going out to forage correspond to the three living habits of crayfish, respectively.

Crayfish belong to ectotherms and are affected by temperature to produce behavioral differences, which range from 20 °C to 35 °C. The temperature is calculated as follows:

where temp represents the temperature of the crayfish's environment.

2.1 Initializing the population

In the d -dimensional optimization problem of COA, each crayfish is a 1 × d matrix representing the solution of the problem. In a set of variables ( X 1 , X 2 , X 3 …… X d ), the position ( X ) of each crayfish is between the upper boundary ( ub ) and lower boundary ( lb ) of the search space. In each evaluation of the algorithm, an optimal solution is calculated, and the solutions calculated in each evaluation are compared, and the optimal solution is found and stored as the optimal solution of the whole problem. The position to initialize the crayfish population is calculated using the following formula.

where X i,j denotes the position of the i-th crayfish in the j-th dimension, ub j denotes the upper bound of the j-th dimension, lb j denotes the lower bound of the j-th dimension, and rand is a random number from 0 to 1.

2.2 Summer escape stage (exploration stage)

In this paper, the temperature of 30 °C is assumed to be the dividing line to judge whether the current living environment is in a high temperature environment. When the temperature is greater than 30 ℃ and it is in the summer, in order to avoid the harm caused by the high temperature environment, crayfish will look for a cool and moist cave and enter the summer to avoid the influence of high temperature. The caverns are calculated as follows.

where X G represents the optimal position obtained so far for this evaluation number, and X L represents the optimal position of the current population.

The behavior of crayfish competing for the cave is a random event. To simulate the random event of crayfish competing for the cave, a random number rand is defined, when rand < 0.5 means that there are no other crayfish currently competing for the cave, and the crayfish will go straight into the cave for the summer. At this point, the crayfish position update calculation formula is as follows.

Here, X new is the next generation position after location update, and C 2 is a decreasing curve. C 2 is calculated as follows.

Here, FEs represents the number of evaluations and MaxFEs represents the maximum number of evaluations.

2.3 Competition stage (exploitation stage)

When the temperature is greater than 30 °C and rand ≥ 0.5, it indicates that the crayfish have other crayfish competing with them for the cave when they search for the cave for summer. At this point, the two crayfish will struggle against the cave, and crayfish X i adjusts its position according to the position of the other crayfish X z . The adjustment position is calculated as follows.

Here, z represents the random individual of the crayfish, and the random individual calculation formula is as follows.

where, N is the population size.

2.4 Foraging stage (exploitation stage)

The foraging behavior of crayfish is affected by temperature, and temperature less than or equal to 30 ℃ is an important condition for crayfish to climb out of the cave to find food. When the temperature is less than or equal to 30 °C, the crayfish will drill out of the cave and judge the location of the food according to the optimal location obtained in this evaluation, so as to find the food to complete the foraging. The position of the food is calculated as follows.

The amount of food crayfish eat depends on the temperature. When the temperature is between 20 °C and 30°C, crayfish have strong foraging behavior, and the most food is found and the maximum food intake is also obtained at 25 °C. Thus, the food intake pattern of crayfish resembles a normal distribution. Food intake was calculated as follows.

Here, µ is the most suitable temperature for crayfish feeding, and σ and C 1 are the parameters used to control the variation of crayfish intake at different temperatures.

The food crayfish get depends not only on the amount of food they eat, but also on the size of the food. If the food is too large, the crayfish can't eat the food directly. They need to tear it up with their claws before eating the food. The size of the food is calculated as follows.

Here, C 3 is the food factor, which represents the largest food, and its value is 3, fitness i represents the fitness value of the i-th crayfish, and fitness food represents the fitness value of the location of the food.

Crayfish use the value of the maximum food Q to judge the size of the food obtained and thus decide the feeding method. When Q > ( C 3 + 1)/2, it means that the food is too large for the crayfish to eat directly, and it needs to tear the food with its claws and eat alternately with the second and third legs. The formula for shredding food is as follows.

After the food is shredded into a size that is easy to eat, the second and third claws are used to pick up the food and put it into the mouth alternately. In order to simulate the process of bipedal eating, the mathematical models of sine function and cosine function are used to simulate the crayfish eating alternately. The formula for crayfish alternating feeding is as follows.

When Q ≤ ( C 3 + 1)/2, it indicates that the food size is suitable for the crayfish to eat directly at this time, and the crayfish will directly move towards the food location and eat directly. The formula for direct crayfish feeding is as follows.

2.5 Pseudo-code for COA

Crayfish optimization algorithm pseudo-code

3 Modified crayfish optimization algorithm (MCOA)

Based on crayfish optimization algorithm, we propose a high-dimensional feature selection problem solving algorithm (MCOA) based on improved crayfish optimization algorithm. In MCOA, we know that the quality of the aquatic environment has a great impact on the survival of crayfish, according to the living habits of crayfish, which mostly feed on plants and like fresh water. Oxygen is an indispensable energy for all plants and animals to maintain life, the higher the content of dissolved oxygen in the water body, the more vigorous the feeding of crayfish, the faster the growth, the less disease, and the faster the water flow in the place of better oxygen permeability, more aquatic plants, suitable for survival, so crayfish has a strong hydrotaxis. When crayfish perceive that the current environment is too dry and hot or lack of food, they crawl backward according to their second, third and foot perception (r) to judge the direction of water flow, and find an aquatic environment with sufficient oxygen and food to sustain life. Good aquatic environment has sufficient oxygen and abundant aquatic plants, to a certain extent, to ensure the survival and reproduction of crayfish.

In addition, we introduce ghost opposition-based learning to help MCOA escape the local optimal trap. The ghost opposition-based learning strategy combines the candidate individual, the current individual and the optimal individual to randomly generate a new candidate position to replace the previous poor candidate position, and then takes the best point or the candidate solution as the central point, and then carries out more specific and extensive exploration of other positions. Traditional opposition-based learning (Mahdavi et al. 2018 ) is based on the central point and carries out opposition-based learning in a fixed format. Most of the points gather near the central point and their positions will not exceed the distance between the current point and the central point, and most solutions will be close to the optimal individual. However, if the optimal individual is not near the current exploration point, the algorithm will fall into local optimal and it is difficult to find the optimal solution. Compared with traditional opposition-based learning, ghost opposition-based learning is a opposition-based learning solution that can be dynamically changed by adjusting the size of parameter k, thereby expanding the algorithm's exploration range of space, effectively solving the problem that the optimal solution is not within the search range based on the center point, and making the algorithm easy to jump out of the local optimal.

According to the life habits of crayfish, this paper proposes a Modified Crayfish Optimization Algorithm (MCOA), which uses environment update mechanism and ghost opposition-based learning strategy to improve COA, and shows the implementation steps, pseudo-code and flow chart of MCOA algorithm as follows.

3.1 Environment update mechanism

In the environmental renewal mechanism, a water quality factor V is introduced to represent the quality of the aquatic environment at the current location. In order to simplify the design and computational complexity of the system, the water quality factor V of the MCOA is represented by a hierarchical discretization, and its value range is set to 0 to 5. Crayfish perceive the quality of the current aquatic environment through the perception ( r ) of the second and third legs, judge whether the current living environment can continue to survive through the perception, and independently choose whether to update the current location. The location update is calculated as follows.

Among them, each crayfish has a certain difference in its own perception of water environment r , X 2 is a random position between the candidate optimal position and the current position, which is calculated by Eq. ( 15 ), X 1 is a random position in the population, and B is an adaptive water flow factor, which is calculated by Eq. ( 16 ).

Among them, the sensing force r of the crayfish’s second and third legs is a random number [0,1]. c is a constant that represents the water flow velocity factor with a value of 2. When V ≤ 3, it indicates that the crayfish perceives the quality of the current living environment to be good and is suitable for continued survival. When V > 3, it indicates that the crayfish perceives that the current living environment quality is poor, and it needs to crawl in the opposite direction according to the direction of water flow that crayfish perceives, so as to find an aquatic environment with sufficient oxygen and abundant food Fig. 1 .

Classification of MAs

In the environmental updating mechanism, in order to describe the behavior of crayfish upstream in more detail, the perception area of crayfish itself is abstractly defined as a circle in MCOA, and crayfish is in the center of the circle. In each evaluation calculation, a random Angle θ is first calculated by the roulette wheel selection algorithm to determine the moving direction of the crayfish in the circular area, and then the moving path of the crayfish is determined according to the current moving direction. In the whole circle, random angles can be chosen from 0 to 360 degrees, from which the value of θ can be determined to be of magnitude [− 1,1]. The difference of random Angle indicates that each crayfish moves its position in a random direction, which broadens the search range of crayfish, enhances the randomness of position and the ability to escape from local optimum, and avoids local convergence Fig. 2 .

Schematic diagram of the environment update mechanism

3.2 Ghost opposition-based learning strategy

The ghost opposition-based learning strategy takes a two-dimensional space as an example. It is assumed that there is a two-dimensional space, as shown in Fig. 3 . On the X-axis, [ ub , lb ] represents the search range of the solution, and the ghost generation method is shown in Fig. 3 . Assuming that the position of a new candidate solution is Xnew and the height of the solution is h1 i , the position of the best solution on the X-axis is the projected position of the candidate solution, and the position and height are XG , h2 i , respectively. In addition, on the X-axis there is a projection position X i of the candidate solution with a height of h3 i. Thus, the position of the ghost is obtained. The projection position of the ghost on the X-axis is x i by vector calculation, and its height is h i . The ghost position is calculated using the following formula.

Schematic diagram of ghost opposition-based learning strategy

In Fig. 3 , the Y-axis represents the convex lens. Suppose there is a ghost position P i , where x i is its projection on the X-axis and h i is its height. P* i is the real image obtained by convex lens imaging. P* i is projected on the X-axis as x* i and has height h* i . Therefore, the opposite individual x* i of individual x i can be obtained. x* i is the corresponding point corresponding to the ghost individual x i obtained from O as the base point. According to the lens imaging principle, we can obtain Eq. ( 18 ), and the calculation formula is as follows.

The strategy formula of ghost opposition-based learning is evolved from Eq. ( 18 ). The strategy formula of ghost opposition-based learning is calculated as follows.

3.3 Implementation of MCOA algorithm

3.3.1 initialization phase.

Initialize the population size N , the population dimension d , and the number of evaluations FEs . The initialized population is shown in Eq. ( 2 ).

3.3.2 Environment update mechanism

Crayfish judge the quality of the current aquatic environment according to the water quality factor V , and speculate whether the current aquatic environment can continue to survive. When V > 3 indicates that the crawfish perceives the quality of the current aquatic environment as poor and is not suitable for survival. According to the sensory information of the second and third legs and the adaptive flow factor, the crawfish judges the direction of the current flow, and then moves upstream to find a better aquatic environment to update the current position. The position update formula is shown in Eq. ( 14 ). When V < 3, it means that the crayfish has a good perception of the current living environment and is suitable for survival, and does not need to update its position.

3.3.3 Exploration phase

When the temperature is greater than 30 ℃ and V > 3, it indicates that crayfish perceive the current aquatic environment quality is poor, and the cave is dry and without moisture, which cannot achieve the effect of summer vacation. It is necessary to first update the position by crawling in the reverse direction according to the flow direction, and find a cool and moist cave in a better quality aquatic environment for summer.

3.3.4 Exploitation stage

When the temperature is less than 30 ℃ and V > 3, it indicates that crayfish perceive the current aquatic environment is poor, and there is not enough food to maintain the survival of crayfish. It is necessary to escape from the current food shortage living environment by crawling in the reverse direction according to the current direction, and find a better aquatic environment to maintain the survival and reproduction of crayfish.

3.3.5 Ghost opposition-based learning strategy

Through the combination of the candidate individual, the current individual and the optimal individual, a candidate solution is randomly generated and compared with the current solution, the better individual solution is retained, the opposite individual is obtained, and the location of the ghost is obtained. The combination of multiple positions effectively prevents the algorithm from falling into local optimum, and the specific implementation formula is shown in Eq. ( 19 ).

3.3.6 Update the location

The position of the update is determined by comparing the fitness values. If the fitness of the current individual update is better, the current individual replaces the original individual. If the fitness of the original individual is better, the original individual is retained to exist as the optimal solution.

The pseudocode for MCOA is as follows (Algorithm 2).

Modified Crayfish optimization algorithm pseudo-code

The flow chart of the MCOA algorithm is as follows.

3.4 Computational complexity analysis

The complexity analysis of algorithms is an essential step to evaluate the performance of algorithms. In the experiment of complexity analysis of the algorithm, we choose the IEEE CEC2020 Special Session and Competition as the complexity evaluation standard of the single objective optimization algorithm. The complexity of MCOA algorithm mainly depends on several important parameters, such as the population size ( N = 30), the number of dimensions of the problem ( d = 10), the maximum number of evaluations of the algorithm ( MaxFEs = 100,000) and the solution function ( C ). Firstly, the running time of the test program is calculated and the running time ( T 0 ) of the test program is recorded, and the test program is shown in Algorithm 3. Secondly, under the same dimension of calculating the running time of the test program, the 10 test functions in the IEEE CEC2020 function set were evaluated 100,000 times, and their running time ( T 1 ) was recorded. Finally, the running time of 100,000 evaluations of 10 test functions performed by MCOA for 5 times under the same dimension was recorded, and the average value was taken as the running time of the algorithm ( T 2 ). Therefore, the formula for calculating the time complexity of MCOA algorithm is given in Eq. ( 21 ).

IEEE CEC2020 complexity analysis test program

The experimental data table of algorithm complexity analysis is shown in Table 1 . In the complexity analysis of the algorithm, we use the method of comparing MCOA algorithm with other seven metaheuristic algorithms to illustrate the complexity of MCOA. In Table 1 , we can see that the complexity of MCOA is much lower than other comparison algorithms such as ROA, STOA, and AOA. However, compared with COA, the complexity of MCOA is slightly higher than that of COA because it takes a certain amount of time to update the location through the environment update mechanism and ghost opposition-based learning strategy. Although the improved strategy of MCOA increases the computation time to a certain extent, the optimization performance of MOCA has been significantly improved through a variety of experiments in section four of this paper, which proves the good effect of the improved strategy.

4 Experimental results and discussion

The experiments are carried out on a 2.50 GHz 11th Gen Intel(R) Core(TM) i7-11,700 CPU with 16 GB memory and 64-bit Windows11 operating system using Matlab R2021a. In order to verify the performance of MCOA algorithm, MCOA is compared with nine metaheuristic algorithms in this subsection. In the experiments, we used the IEEE CEC2020 test function to evaluate the optimization performance of the MCOA algorithm Fig. 4 .

Flow chart of the MCOA algorithm

4.1 Experiments with IEEE CEC2020 test functions

In this subsection, using the Crayfish Optimization Algorithm (COA), Remora Optimization Algorithm (ROA) (Jia et al. 2021 ), Sooty Tern Optimization Algorithm (STOA) (Dhiman and Kaur 2019 ), Arithmetic Optimization Algorithm (AOA) (Abualigah et al. 2021 ), Harris Hawk Optimization Algorithm (HHO) (Heidari et al. 2019 ), Prairie Dog Optimization Algorithm (PDO) (Ezugwu et al. 2022 ), Genetic Algorithm (GA) (Mirjalili and Mirjalili 2019 ),Modified Sand Cat Swarm Optimization Algorithm (MSCSO) (Wu et al. 2022 ) and a competition algorithm LSHADE (Piotrowski 2018 ) were compared to verify the optimization effect of MCOA. The parameter Settings of each algorithm are shown in Table 2 .

In order to test the performance of MCOA, this paper selects 10 benchmark test functions of IEEE CEC2020 for simulation experiments. Where F1 is a unimodal function, F2–F3 is a multimodal function, F4 is a non-peak function, F5–F7 is a hybrid function, and F8-F10 is a composite function. The parameters of this experiment are uniformly set as follows: the maximum number of evaluation MaxFEs is 100,000, the population size N is 30, and the dimension size d is 10. The MCOA algorithm and the other nine algorithms are run independently for 30 times, and the average fitness value, standard deviation of fitness value and Friedman ranking calculation of each algorithm are obtained. The specific function Settings of the IEEE CEC2020 benchmark functions are shown in Table 3 .

4.1.1 Results statistics and convergence curve analysis of IEEE CEC2020 benchmark functions

In order to more clearly and intuitively compare the ability of MCOA and various algorithms to find individual optimal solutions, the average fitness value, standard deviation of fitness value and Friedman ranking obtained by running MCOA and other comparison algorithms independently for 30 times are presented in the form of tables and images. The data and images are shown in Table 4 and Fig. 5 respectively.

Convergence curve of MCOA algorithm in IEEE CEC2020

In Table 4 , mean represents the average fitness value, std represents the standard deviation of fitness value, rank represents the Friedman ranking, Friedman average rank represents the average ranking of the algorithm among all functions, and Friedman rank represents the final ranking of this algorithm. Compared with other algorithms, MCOA achieved the best results in average fitness value, standard deviation of fitness value and Friedman ranking. In unimodal function F1, although MCOA algorithm is slightly worse than LSHADE algorithm, MCOA is superior to other algorithms in mean fitness value, standard deviation of fitness value, Friedman ranking and other aspects. In the multimodal functions F2 and F3, although the average fitness value of MCOA is slightly worse, it also achieves a good result of ranking second. The standard deviation of fitness value in F3 is better than other comparison algorithms in terms of stability. In the peakless function F4, except GA and LSHADE algorithm, other algorithms can find the optimal individual solution stably. In the mixed functions F5, F6, and F7, although the mean fitness value of LSHADE is better than that of MCOA, the standard deviation of the fitness value of MCOA is better than that of the other algorithms compared. Among the composite functions of F8, F9 and F10, the standard deviation of MCOA's fitness value at F8 is slightly worse than that of LSHADE, but the average fitness value and standard deviation of fitness value are the best in other composite functions, and it has achieved the first place in all composite functions. Finally, from the perspective of Friedman average rank, MCOA has a strong comprehensive performance and still ranks first. Through the analysis of the data in Table 4 , it can be seen that MCOA ranks first overall and has good optimization effect, and its optimization performance is better than other 9 comparison algorithms.

Figure 5 shows that in the IEEE CEC2020 benchmark functions, for the unimodal function F1, although LSHADE algorithm has a better optimization effect, compared with similar meta-heuristic algorithms, MCOA has a slower convergence rate in the early stage, but can be separated from local optimal and converge quickly in the middle stage. In the multimodal functions F2 and F3, similar to F1, MCOA converges faster in the middle and late stages, effectively exiting the local optimal. Although the convergence speed is slower than that of LSHADE, the optimal value can still be found. In the peak-free function F4, the optimal value can be found faster by all algorithms except LSHADE, STOA and PDO because the function is easy to implement. In the mixed functions F5, F6 and F7, although the convergence rate of MCOA is slightly slower than that of COA algorithm in the early stage, it can still find better values than the other eight algorithms except LSHADE in the later stage. For the composite functions F8, F9 and F10, MCOA can find the optimal value faster than the other nine algorithms.

Based on the above, although LSHADE has a stronger ability to find the optimal value in a small number of functions, MCOA can still find the optimal value in most functions in the later stage, and compared with the other eight pair algorithms of the same type, MCOA has more obvious enhancement in optimization ability and avoidance of local optimization, and has better application effect.

4.1.2 Analysis of Wilcoxon rank sum test results

In the comparison experiment, the different effects of multiple algorithms solving the same problem are used to judge whether each algorithm has high efficiency and more obvious influence on solving the current problem, such as the convergence speed of the convergence curve, the fitness value of the optimal solution, the ability to jump out of the local optimum, etc. At present, only the average fitness value, the standard deviation of fitness value and the convergence curve can not be used as the basis for judging whether the performance of the algorithm is efficient. Therefore, the data and images presented by each algorithm in solving the current problem are comprehensively analyzed, and the Wilcoxon rank sum test is used to further verify the difference between MCOA and the other nine comparison algorithms. In this experiment, the significance level is defined as 5%. If its calculated value is less than 5%, it proves that there is a significant difference between the two algorithms, and if it is greater than 5%, it proves that there is no significant difference between the two algorithms. Table 5 shows the Wilcoxon rank-sum test results of the MCOA algorithm and the other nine comparison algorithms. Where the symbols “ + ”, “−” and “ = ” table the performance of MCOA better, worse and equal to the comparison algorithms, respectively.

In the calculation of the function F4 without peak, the value of 1 appears in the comparison of various algorithms such as MCOA, COA, ROA, STOA and other algorithms, indicating that in this function, a variety of algorithms have found the optimal value, there is no obvious difference, which can be ignored. However, in most of the remaining functions, the significance level of MCOA compared with the other nine algorithms is less than 5%, which is a significant difference.

From the overall table, the MCOA algorithm also achieves good results in the Wilcoxon rank-sum test of the IEEE CEC2020 benchmark function, and the contrast ratio with other algorithms is less than 5%, which proves that the MCOA algorithm has a significant difference from the other nine algorithms, and MCOA has better optimization performance. According to the comparison results with the original algorithm, it is proved that MCOA algorithm has a good improvement effect.

4.2 Comparison experiment of single strategy

MCOA adopts two strategies, environment update mechanism and ghost opposition-based learning strategy, to improve COA. In order to prove the effectiveness of these two strategies for algorithm performance optimization, a single strategy comparison experiment is added in this section. In the experiment in this section, EUCOA algorithm which only adds environment update mechanism and GOBLCOA algorithm which only adds ghost opposition-based learning strategy are compared with the basic COA algorithm. The experiments are independently run 30 times in IEEE CEC2020 benchmark test function, and the statistical data obtained are shown in Table 6 . In order to make the table easy to view the statistical results, the poor data in the table will be bolded to make the statistical results more clear and intuitive. It can be seen from the table that among the best fitness values, average fitness values and standard deviation of fitness values of the 10 test functions, GOBLCOA and EUCOA account for less bolded data, while most data of the original algorithm COA are bolded in the table, which effectively proves that both the environment update mechanism and the ghost opposition-based learning strategy play a certain role in COA. The comprehensive performance of COA has been significantly improved.

4.3 Parameter sensitivity analysis of water flow velocity factor c

In order to better prove the influence of flow velocity coefficient on MCOA, we choose different flow velocity coefficient c values for comparison experiments. Table 7 shows the statistical results of 30 independent runs of different water flow velocity coefficients in CEC2020. The bold sections represent the best results. As can be seen from the table, the result obtained by c = 2 is significantly better than the other values. Only in individual test functions are the results slightly worse. In the F1 function, c = 5 has the best std. In the F5 function, std is best at c = 6. Among F10 functions, c = 5 has the best std. Among the other test functions, both the mean fitness value and std at water flow velocity factor c = 2 are optimal. Through the above analysis, it is proved that the water flow velocity factor c = 2 has a good optimization effect.

4.4 Experimental summary

In this section, we first test MCOA's optimization performance on the IEEE CEC2020 benchmark function. The improved MCOA is compared with the original algorithm COA and six other meta-heuristic algorithms in the same environment and the experimental analysis is carried out. Secondly, the rank sum test is used to verify whether there are significant differences between MCOA and the other nine comparison algorithms. Finally, three algorithms, EUCOA with environment update mechanism, GOBLCOA with ghost opposition-based learning strategy, COA and MCOA, are tested to improve performance. These three experimental results show that MCOA has a good ability to find optimal solutions and get rid of local optimal solutions.

5 Constrained engineering design problems

With the new development of the era of big data, the solution process becomes complicated and the calculation results become accurate, and more and more people pay close attention to the dynamic development of the feasibility and practicality of the algorithm, so as to ensure that the algorithm has good practical performance on constrained engineering design problems. In order to verify the optimization effect of MCOA in practical applications, four constrained engineering design problems are selected for application testing of MCOA to evaluate the performance of MCOA in solving practical application problems. Every constrained engineering design problems has a minimization objective function (Papaioannou and Koulocheris 2018 ) that is used to calculate the fitness value for a given problem. In addition, each problem contains a varying number of constraints that are taken into account during the calculation of the objective function. If the constraints are not met, the penalty function (Yeniay 2005 ) is used to adjust the fitness value. However, the processing of constraints is not the focus of our research, our focus is on the optimization of parameters in a convex region composed of constraints (Liu and Lu 2014 ). In order to ensure the fairness of the experiment, the parameters of all experiments in this section are set as follows: the maximum evaluation time MaxFEs is 10,000 and the overall scale N is 30. In each experiment, all the algorithms were analyzed 500 times and the optimal results were obtained.

5.1 Multi-disc clutch braking problem

In the field of vehicle engineering, there is a common constrained engineering design problems multi-disc clutch braking problem, and the purpose of our algorithm is to minimize the mass of the multi-disc clutch by optimizing eight constraints and five variables, so as to improve the performance of the multi-disc clutch. Among them, the five variables are: inner diameter r i , outer diameter r o , brake disc thickness t , driving force F , and surface friction coefficient Z . The specific structure of the multi-disc clutch is shown in Fig. 6 .

Schematic diagram of the multi-disc clutch braking problem

The mathematical formulation of the multi-disc clutch braking problem is as follows.

Objective function:

Subject to:

Variable range:

Other parameters:

After calculation and experiments, the experimental results of the multi-disc clutch braking problem are made into a table as shown in Table 8 . In Table 8 , MCOA concluded that the inner diameter r i = 70, the outer diameter r 0 = 90, the thickness of the brake disc t = 1, the driving force F = 600, and the surface friction coefficient Z = 2. At this time, the minimum weight obtained is 0.2352424, it is 11.16% higher than the original algorithm. Compared with MCOA, the other five algorithms in the calculation of this problem show that the optimization effect is far lower than that of MCOA.

5.2 Design problem of welding beam

The welded beam design problem is very common in the field of structural engineering and is constrained not only by four decision variables (welding width h , connecting beam length l , beam height t , and connecting beam thickness b ) but also by seven other different conditions. Therefore, it is challenging to solve this problem. The purpose of the optimization algorithm is to achieve the best structural performance of the welded beam and reduce its weight by optimizing the small problems such as the shape, size and layout of the weld under many constraints. The specific structure of the welded beam is shown in Fig. 7 .

Schematic diagram of the welded beam design problem

The mathematical formulation of the welded beam design problem is as follows.

Boundaries:

The experimental results of the welding beam design problem are shown in Table 9 . In the table, the welding width obtained by the MCOA algorithm h = 0.203034,the length of the connecting beam is l = 3.310032, the height of the beam is t = 9.084002, and the thickness of the connecting beam is b = 0.20578751. At this time, the minimum weight is 1.707524, it is 1.46% higher than the original algorithm. In the welding beam design problem, the weight determines the application effect of the algorithm in the practical problem. The weight of MCOA algorithm is smaller than that of other algorithms. Therefore, the practical application effect of MCOA is much greater than that of other algorithms.

5.3 Design problem of reducer

A reducer is a mechanical device used to reduce the speed of rotation and increase the torque. Among them, gears and bearings are an indispensable part of the reducer design, which have a great impact on the transmission efficiency, running stability and service life of the reducer. The weight of the reducer also determines the use of the reducer. Therefore, we will adjust the number of teeth, shape, radius and other parameters of the gear in the reducer to maximize the role of the reducer, reduce the friction between the parts, and extend the service life of the reducer. In this problem, a total of seven variables are constrained, which are the gear width x 1 , the gear modulus x 2 , the gear teeth x 3 , the length of the first axis between bearings x 4 , the length of the second axis between bearings x 5 , the diameter of the first axis x 6 and the diameter of the second axis x 7 . The specific structure of the reducer is shown in Fig. 8 .

Schematic diagram of the reducer design problem

The mathematical model of the reducer design problem is as follows.

The experimental results of the reducer design problem are shown in Table 10 . From Table 10 , it is known that the gear width calculated by the MCOA algorithm is x 1 = 3.47635, the gear modulus x 2 = 0.7, the gear teeth x 3 = 17, the length of the first axis between the bearings x 4 = 7.3, the length of the second axis between the bearings × 5 = 7.8, and the length of the first axis between the bearings x 5 = 7.8. The diameter of the first axis is x 6 = 3.348620, the diameter of the second axis is x 7 = 5.2768, and the minimum weight is 2988.27135, it is 0.08% higher than the original algorithm. In this experiment, it can be concluded that MCOA has the smallest data among the minimum weights obtained by MCOA and other comparison algorithms in this problem, which proves that MCOA has the best optimization effect in solving such problems.

5.4 Design problem of three-bar truss

Three-bar truss structure is widely used in bridge, building, and mechanical equipment and other fields. However, the size, shape and connection mode of the rod need to be further explored by human beings. Therefore, A 1 = x 1 and A 2 = x 2 determined by the pairwise property of the system need to be considered in solving this problem. In addition to this, there will be constraints on the total support load, material cost, and other conditions such as cross-sectional area. The structural diagram of the three-bar truss is shown in Fig. 9 .

Schematic diagram of the three-bar truss design problem

The mathematical formulation of the three-bar truss design problem is as follows.

The experimental results of the three-bar truss design problem are shown in Table 11 , from which it can be concluded that x 1 = 0.7887564and x 2 = 0.4079948of the MCOA algorithm on the three-bar truss design problem. At this time, the minimum weight value is 263.85438633, it is 0.24% higher than the original algorithm. Compared with the minimum weight value of other algorithms, the value of MCOA is the smallest. It is concluded that the MCOA algorithm has a good optimization effect on the three-bar truss design problem.

The experimental results of four constrained engineering design problems show that MCOA has good optimization performance in dealing with problems similar to constrained engineering design problems. In addition, we will also introduce the high-dimensional feature selection problem of the wrapper method, and further judge whether MCOA has good optimization performance and the ability to deal with diversified problems through the classification and processing effect of data.

6 High-dimensional feature selection problem

The objective of feature selection is to eliminate redundant and irrelevant features, thereby obtaining a more accurate model. However, in high-dimensional feature spaces, feature selection encounters challenges such as high computational costs and susceptibility to over-fitting. To tackle these issues, this paper propose novel high-dimensional feature selection methods based on metaheuristic algorithms. These methods aim to enhance the efficiency and effectiveness of feature selection in complex, high-dimensional datasets.

High-dimensional feature selection, as discussed in reference (Ghaemi and Feizi-Derakhshi 2016 ), focuses on processing high-dimensional data to extract relevant features while eliminating redundant and irrelevant ones. This process enhances the model's generalization ability and reduces computational costs. The problem of high-dimensional feature selection is often referred to as sparse modeling, encompassing two primary methods: filter and wrapper. Filter methods, also called classifier-independent methods, can be categorized into univariate and multivariate methods. Univariate methods consider individual features independently, leveraging the correlation and dependence within the data to quickly screen and identify the optimal feature subset. On the other hand, multivariate methods assess relationships between multiple features simultaneously, aiming to comprehensively select the most informative feature combinations. Wrapper methods offer more diverse solutions. This approach treats feature selection as an optimization problem, utilizing specific performance measures of classifiers and objective functions. Wrapper methods continuously explore and evaluate various feature combinations to find the best set of features that maximizes the model’s performance. Unlike filter methods, wrapper methods provide a more customized and problem-specific approach to feature selection.

Filter methods, being relatively single and one-sided, approach the problem of feature selection in a straightforward manner by considering individual features and their relationships within the dataset. However, they might lack the flexibility needed for complex and specific problem scenarios. However, wrapper methods offer tailored and problem-specific solutions. They exhibit strong adaptability, wide applicability, and high relevance to the specific problem at hand. Wrapper methods can be seamlessly integrated into any learning algorithm, allowing for a more customized and targeted approach to feature selection. By treating feature selection as an optimization problem and continuously evaluating different feature combinations, wrapper methods can maximize the effectiveness of the algorithm and optimize its performance to a greater extent compared to filter methods. In summary, wrapper methods provide a more sophisticated and problem-specific approach to feature selection, enabling the algorithm to achieve its maximum potential by selecting the most relevant and informative features for the given task.

6.1 Fitness function

In this subsection, the wrapper method in high-dimensional feature selection is elucidated, employing the classification error rate (CEE) (Wang et al. 2005 ) as an illustrative example. CEE is utilized as the fitness function or objective function to assess the optimization effectiveness of the feature selection algorithm for the problem at hand. Specifically, CEE quantifies the classification error rate when employing the k-nearest-neighbors (KNN) algorithm (Datasets | Feature Selection @ ASU. 2019 ), with the Euclidean distance (ED) (The UCI Machine Learning Repository xxxx) serving as the metric for measuring the distance between the current model being tested and its neighboring models. By using CEE as the fitness function, the wrapper method evaluates different feature subsets based on their performance in the context of the KNN algorithm. This approach enables the algorithm to identify the most relevant features that lead to the lowest classification error rate, thereby optimizing the model's performance. By focusing on the accuracy of classification in a specific algorithmic context, the wrapper method ensures that the selected features are highly tailored to the problem and the chosen learning algorithm. This targeted feature selection process enhances the overall performance and effectiveness of the algorithm in handling high-dimensional data.

where X denotes feat, Y denotes label, both X and Y are specific features in the given data model, and D is the total number of features recorded.

In the experimental setup, each dataset is partitioned into a training set and a test set, with an 80% and 20% ratio. The training set is initially utilized to select the most characteristic features and fine-tune the parameters of the KNN model. Subsequently, the test set is employed to evaluate and calculate the data model and algorithm performance. To address concerns related to fitting ability and overfitting, hierarchical cross-validation with K = 10 was employed in this experiment. In hierarchical cross-validation, the training portion is divided into ten equal-sized subsets. The KNN classifier is trained using 9 out of the 10 folds (K-1 folds) to identify the optimal KNN classifier, while the remaining fold is used for validation purposes. This process is repeated 10 times, ensuring that each subset serves both as a validation set and as part of the training data. This iterative approach is a crucial component of our evaluation methodology, providing a robust assessment of the algorithm's performance. By repeatedly employing replacement validation and folding training, we enhance the reliability and accuracy of our evaluation, enabling a comprehensive analysis of the algorithm's effectiveness across various datasets.

6.2 High-dimensional datasets

In this subsection, the optimization performance of MCOA is assessed using 12 high-dimensional datasets sourced from the Arizona State University (Too et al. 2021 ) and University of California Irvine (UCI) Machine Learning databases (Chandrashekar and Sahin 2014 ). By conducting experiments on these high-dimensional datasets, the results obtained are not only convincing but also pose significant challenges. These datasets authentically capture the intricacies of real-life spatial problems, making the experiments more meaningful and applicable to complex and varied spatial scenarios. For a detailed overview of the 12 high-dimensional datasets, please refer to Table 12 .

6.3 Experimental results and analysis

In order to assess the effectiveness and efficiency of MCOA in feature selection, we conducted comparative tests using MCOA as well as several other algorithms including COA, SSA, PSO, ABC, WSA (Baykasoğlu et al. 2020 ), FPA (Yang 2012 ), and ABO (Qi et al. 2017 ) on 12 datasets. In this section of the experiment, the fitness value of each algorithm was calculated, and the convergence curve, feature selection accuracy (FS Accuracy), and selected feature size for each algorithm were analyzed. Figures 10 , 11 and 12 display the feature selection (FS) convergence curve, FS Accuracy, and selected feature size for the eight algorithms across the 12 datasets. From these figures, it is evident that the optimization ability and prediction accuracy of the MCOA algorithm surpass those of the other seven comparison algorithms. Taking the dataset CLL-SUB-111 as an example in Figs. 11 and 12 , MCOA selected 20 features, while the other seven algorithms selected more than 2000 features. Moreover, the prediction accuracy achieved by MCOA was higher than that of the other seven algorithms. Across all 12 datasets, the comparison figures indicate that the MCOA algorithm consistently outperforms the others. Specifically, the MCOA algorithm tends to select smaller feature subsets, leading to higher prediction accuracy and stronger optimization capabilities. This pattern highlights the superior performance of MCOA in feature selection, demonstrating its effectiveness in optimizing feature subsets for improved prediction accuracy.

Convergence curve of FS

Comparison plot of verification accuracy of eight algorithms

Comparison plots of feature sizes of the eight algorithms

To address the randomness and instability inherent in experiments, a single experiment may not fully demonstrate the effectiveness of algorithm performance. Therefore, we conducted 30 independent experiments using 12 datasets and 8 algorithms. For each algorithm and dataset combination, we calculated the average fitness value, standard deviation of the fitness value, and Friedman rank. Subsequently, the Wilcoxon rank sum test was employed to determine significant differences between the performance of different algorithms across various datasets. Throughout the experiment, a fixed population size of 10 and a maximum of 100 iterations were used. The 12 datasets were utilized to evaluate the 8 algorithms 300 times (tenfold cross-validation × 30 runs). It is essential to note that all algorithms were assessed using the same fitness function derived from the dataset, ensuring a consistent evaluation criterion across the experiments. By conducting multiple independent experiments and statistical analyses, the study aimed to provide a comprehensive and robust assessment of algorithm performance. This approach helps in drawing reliable conclusions regarding the comparative effectiveness of the algorithms under consideration across different datasets, accounting for the inherent variability and randomness in the experimental process.

Table 13 presents the average fitness calculation results from 30 independent experiments for the eight algorithms, it is 55.23% higher than the original algorithm. According to the table, in the Ionosphere dataset, MCOA exhibits the best average fitness, albeit with slightly lower stability compared to ABC. Similarly, in the WarpAR10P dataset, MCOA achieves the best average fitness, with stability slightly lower than COA. After conducting Friedman ranking on the fitness calculation results of the 30 independent experiments, it is concluded that although MCOA shows slightly lower stability in some datasets, it ranks first overall. Among the other seven algorithms, PSO ranks second, ABO ranks third, COA ranks fourth, and ABC, SSA, FPA, and WSA rank fifth to ninth, respectively. These results demonstrate that MCOA exhibits robust optimization performance and high stability in solving high-dimensional feature selection problems. Moreover, MCOA outperforms COA, showcasing its superior improvement in solving these complex problems.

Table 14 presents the accuracy calculation results of the eight algorithms for 30 independent experiments, it is 10.85% higher than the original algorithm. According to the table, the average accuracy of MCOA is the highest across all datasets. Notably, in the Colon dataset, MCOA performs exceptionally well with a perfect average accuracy of 100%. However, in the Ionosphere dataset, MCOA exhibits slightly lower stability compared to ABC, and in the WarpAR10P dataset, it is slightly less stable than COA. Upon conducting Friedman ranking on the average accuracy calculation results of 30 independent experiments, it is evident that MCOA ranks first overall. Among the other seven algorithms, PSO ranks second, ABC ranks third, COA ranks fourth, and ABO, FPA, SSA, and WSA rank fifth to ninth, respectively. These results highlight that MCOA consistently achieves high accuracy and stability in solving high-dimensional feature selection problems. Its superior performance across various datasets underscores its effectiveness and reliability in real-world applications.

Table 15 demonstrates that the MCOA algorithm has shown significant results in the Wilcoxon rank sum test for high-dimensional feature selection fitness. The comparison values with other algorithms are less than 5%, indicating that the MCOA algorithm exhibits significant differences compared to the other seven algorithms. This result serves as evidence that MCOA outperforms the other algorithms, showcasing its superior optimization performance. Additionally, when comparing the results with the original algorithm, it becomes evident that the MCOA algorithm has a substantial and positive impact, demonstrating its effectiveness and improvement over existing methods. These findings underscore the algorithm's potential and its ability to provide substantial enhancements in the field of high-dimensional feature selection.

7 Conclusions and future work

The Crayfish Optimization Algorithm (COA) is grounded in swarm intelligence, drawing inspiration from crayfish behavior to find optimal solutions within a specific range. However, COA’s limitations stem from neglecting crucial survival traits of crayfish, such as crawling against water to discover better aquatic environments. This oversight weakens COA’s search ability, making it susceptible to local optima and hindering its capacity to find optimal solutions. To address these issues, this paper introduces a Modified Crayfish Optimization Algorithm (MCOA). MCOA incorporates an environmental updating mechanism, enabling crayfish to randomly select directions toward better aquatic environments for location updates, enhancing search ability. The addition of the ghost opposition-based learning strategy expands MCOA’s search range and promotes escape from local optima. Experimental validations using IEEE CEC2020 benchmark functions confirm MCOA’s outstanding optimization performance.

Moreover, MCOA’s practical applicability is demonstrated through applications to four constrained engineering problems and high-dimensional feature selection challenges. These experiments underscore MCOA’s efficacy in real-world scenarios, but MCOA can only solve the optimization problem of a single goal. In future studies, efforts will be made to further optimize MCOA and enhance its function. We will exploitation multi-objective version of the algorithm to increase the search ability and convergence of the algorithm through non-dominated sorting, multi-objective selection, crossover and mutation, etc., to solve more complex practical problems. It is extended to wireless sensor network coverage, machine learning, image segmentation and other practical applications.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to thank the support of Fujian Key Lab of Agriculture IOT Application, IOT Application Engineering Research Center of Fujian Province Colleges and Universities, Guiding Science and Technology Projects in Sanming City (2023-G-5), Industry-University Cooperation Project of Fujian Province (2021H6039), Fujian Province Industrial Guidance (Key) Project (2022H0053), Sanming Major Science and Technology Project of Industry-University-Research Collaborative Innovation (2022-G-4), and also the anonymous reviewers and the editor for their careful reviews and constructive suggestions to help us improve the quality of this paper.

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Jia, H., Zhou, X., Zhang, J. et al. Modified crayfish optimization algorithm for solving multiple engineering application problems. Artif Intell Rev 57 , 127 (2024). https://doi.org/10.1007/s10462-024-10738-x

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Two multi-ethnic workers working in a plastics factory, standing on the factory floor, looking at ... [+] the control panel of one of the machines. The African-American man is pointing to the panel. His coworker, an Hispanic woman, is holding a digital tablet.

There is no doubt that today’s business challenges are more complex and global than ever, but I still see my peers and business leaders using the same strategies that worked for them years ago. Aspiring new business owners often sink millions into innovations and marketing plans that never get off the ground, and overlook simple details that cost them time, energy, and success.

For example, many businesses are currently struggling with getting their employees back to the office for work, to improve business productivity, accountability, and customer satisfaction. In fact, this challenge clearly has personal team considerations, as well as business implications. Many people prefer the flexibility and comfort of working from home, outweighing results and growth.

I’m not sure if the real problem here is business process or people management, or both, but there is certainly much room for error on both sides. As a consultant, I found some good strategies for not solving the wrong problem in a recent book, “ Solve the Real Problem ,” by Roger L. Firestien, PhD., from Buffalo State University, Innovation Resources, and other roles.

He has real credentials in academia, as well as problem-solving and innovation experiences with many businesses around the world. He offers some key recommendations that I also espouse for how to zero in on the root challenge and not waste large amounts of time and money you cannot afford:

1. Creative questions are key to problem definition. Focus on chains of fact-finding questions and judgement or decision questions to bring out solution ideas. In all cases, defer judgment and avoid excuses like “I don’t have time.” One good question can generate whole new fields of inquiry and can prompt changes in entrenched thinking.

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Creative questions are also useful for exploring new business ideas. Just remember that solving customer problems is the challenge here, rather than internal problems. The process and the results are the same – starting with creative questions to find the real opportunity.

2. Adopt a more effective problem-solving mindset. Analyze your habitual approach to problem solving and be prepared to challenge your own assumptions. Avoid settling for symptoms as the problem or jumping to conclusions based on poor information or your own biases. Sometimes we get in our own way and end up working on the wrong thing.

This strategy also applies to new opportunities for customer growth as well as organizational problems. I still see too many technologists whose mindset is focused on the beauty of their innovation, rather than the problem it solves for customers.

3. Don’t trust or act on your first impression. We all make wrong judgments on first impressions, especially with recurring problems or with people who are of a different nationality, race, and ethnicity. First impressions are usually wrong, especially if they are made in an emotional environment, under time constraints, or with too little information.

4. Get an outside perspective with no agenda. The best way to get an outside perspective is to tap into people who run in circles different from your own. Look for “creative catalysts” who can provide a fresh perspective on the problem and potential solutions. Beware of experts in the relevant technology who may have their own biases.

5. Look for the bigger picture, not minutiae. Make sure that you don’t become unable to see the “forest for the trees” by looking only at a few details of the problem. Consciously step back and take a broader view of the challenge ahead. This approach also builds alignment with related perspectives and issues, and results in better long-term solutions.

In the real world, my experience is that none of these strategies will work without conscientious business leadership, committed team members, a positive business model, and a viable customer opportunity. Your team also needs the creativity skills and training to properly diagnose problems and challenges, generate solutions, and put these solutions into action.

I encourage all of you to recognize that every business in today’s world will encounter challenges and world-class problems. Thus it behooves all of us to continuously update our business problem-solving strategies, support a culture of innovation, and keep moving forward in your quest to make the world a better place, and enjoy the journey to get there.

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These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, and so the following techniques should reinforce skills you have already begun to develop. Problem-Solving Strategy for Newton's Laws of Motion. Step 1.
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1.7 Solving Problems in Physics
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These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, and so the following techniques should reinforce skills you have already begun to develop. Problem-Solving Strategy for Newton's Laws of Motion. Step 1.
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This document aims to expose you to the process. Solving a physics problem usually breaks down into three stages: Design a strategy. Execute that strategy. Check the resulting answer. This document treats each of these three elements in turn, and concludes with a summary.
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This page titled 1.8: Solving Problems in Physics is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax. The three stages of the process for solving physics problems used in this textmap are as follows: 1)Strategy: Determine which physical principles are involved and develop a strategy for using them to ….
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Furthermore, as we discuss below, the strategies we observe these students using are markers that they have progressed beyond the phase of being complete novices, and have developed some limited physics expertise. Physics 41E covers static equilibrium, forces, torques, 1D kinematics, and conservation of energy.
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Crayfish Optimization Algorithm (COA) is innovative and easy to implement, but the crayfish search efficiency decreases in the later stage of the algorithm, and the algorithm is easy to fall into local optimum. To solve these problems, this paper proposes an modified crayfish optimization algorithm (MCOA). Based on the survival habits of crayfish, MCOA proposes an environmental renewal ...
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The process and the results are the same - starting with creative questions to find the real opportunity. 2. Adopt a more effective problem-solving mindset. Analyze your habitual approach to ...