problem solving and physics

v = 0 m/s

v = 88.3 m/s

( 112 m/s) 2 = (0 m/s) 2 + 2*(a)*(398 m)

12544 m 2 /s 2 = 0 m 2 /s 2 + (796 m)*a

12544 m 2 /s 2 = (796 m)*a

(12544 m 2 /s 2 )/(796 m) = a

a = 15.8 m/s 2

Return to Problem 19

v f 2 = v i 2 + 2*a*d

(0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(91.5 m)

0 m 2 /s 2 = v i 2 - 1793 m 2 /s 2

1793 m 2 /s 2 = v i 2

v i = 42.3 m/s

Now convert from m/s to mi/hr:

v i = 42.3 m/s * (2.23 mi/hr)/(1 m/s)

v i = 94.4 mi/hr

Return to Problem 20

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.

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.

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction

• Authors: William Moebs, Samuel J. Ling, Jeff Sanny
• Publisher/website: OpenStax
• Book title: University Physics Volume 1
• Publication date: Sep 19, 2016
• Location: Houston, Texas
• Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction
• Section URL: https://openstax.org/books/university-physics-volume-1/pages/1-7-solving-problems-in-physics

© Jan 19, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

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.

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]

Privacy Policy

Collection of Solved Problems in Physics

Welcome in collection of solved problems in physics.

This collection of Solved Problems in Physics is developed by Department of Physics Education , Faculty of Mathematics and Physics , Charles University in Prague since 2006.

We are looking for authors of tasks in English or other languages. The database interface is prepared to be run in any language. If you are interested please contact administrator at sbirka@kdf.mff.cuni.cz .

The Collection contains tasks at various level in mechanics, electromagnetism, thermodynamics and optics. The tasks mostly do not contain the calculus (derivations and integrals), on the other hand they are not simple such as the use just one formula. Tasks have very detailed solution descriptions. Try to solve the task or at least some of its parts independently. You can use hints and task analysis (solution strategy).

Level and category of the task is indicated by an icon in the right upper corner.

Tasks in English are translated from the Czech part of the Collection.The development of the Colection interface as well as translations of tasks was supported by projects FRVŠ (759 F6d/2008, 788 F6d/2010, FRVŠ 888 F6d/2011) a by IRP.

In case of problems or ideas how to improve the Collection please contact administrator .

45 Users Online

Active students, messages sent, images uploaded, free learning, end of free trial.

Unlock faster, more accurate responses + 20 more PRO features.

Phy Pro Trial -

Free but limited access to Phy Pro. Need more power?  

NEW - Bookmark Phy.Chat

Need to access this page fast? Just type in Phy.Chat into google.

Snap a picture of the problem straight from your phone.

Smart Options

Phy automatically generates short follow ups. Just click it.

New - Learning Lab

Customize your learning to help Phy adapt to you even quicker.

Coming Soon - Chat Notes

View all chats with Phy, save to notes, & create study guides.

Phy Adaptive Engine®

The more you solve, the better Phy adapts to your learning style. 

Learn it. Solve it. Grade it. Explain it. With Phy.

Free Response Question? Upload a image of your working. Phy will grade it. 

Teacher didn’t explain it? Take a picture of the board and give it to Phy. 

Can’t solve a problem? Phy can. And it will show you the best approach. 

Upgrade to Phy Pro.

Phy Version 8 (3.20.24) - Systems Operational

The most advanced version of Phy. Currently 50% off, for early supporters.

Billed Monthly. Cancel Anytime.

Trial   –>  Phy Pro

• Unlimited Messages
• Unlimited Image Uploads
• Unlimited Smart Actions
• Unlimited UBQ Credits
• 30 --> 300 Word Input
• 3 --> 15 MB Image Size Limit
• 1 --> 3 Images per Message
• 200% Memory Boost
• 150% Better than GPT
• 75% More Accurate, 50% Faster
• Mobile Snaps

Prof Phy ULTRA

Access will be given out on a rolling basis. You must have an active Phy Pro subscription, for at least 90 days, to automatically join the waitlist. 

Features include:

• Save To Notes
• Personalize Phy
• Smart Actions V2
• Instant Responses
• Phy Adaptive Engine

Share Phy.Chat

Enjoying Phy? Share the 🔗 with friends!

Welcome to Phy Panel.

Here you can customize Phy to your preferences. Currently available to only Ultra users. Pro users will get access on a rolling basis. 

Not currently eligible to use Phy Panel.

Report a bug.

What went wrong? 

You must be signed in to leave feedback

Discover the world's best Physics resources

Continue with.

By continuing you (1) agree to our Terms of Sale  and Terms of Use and (2) consent to sharing your IP and browser information used by this site’s security protocols as outlined in our Privacy Policy .

• Ask yourself if your answers make sense. If the numbers look absurd (for example, you get that a rock dropped off a 50-meter cliff moves with the speed of only 0.00965 meters per second when it hits the ground), you made a mistake somewhere.
• Don't forget to include the units into your answers, and always keep track of them. So, if you are solving for velocity and get your answer in seconds, that is a sign that something went wrong, because it should be in meters per second.
• Plug your answers back into the original equations to make sure you get the same number on both sides.

Community Q&A

• Many people report that if they leave a problem for a while and come back to it later, they find they have a new perspective on it and can sometimes see an easy way to the answer that they did not notice before. Thanks Helpful 249 Not Helpful 48
• Try to understand the problem first. Thanks Helpful 186 Not Helpful 51
• Remember, the physics part of the problem is figuring out what you are solving for, drawing the diagram, and remembering the formulae. The rest is just use of algebra, trigonometry, and/or calculus, depending on the difficulty of your course. Thanks Helpful 115 Not Helpful 34

• Physics is not easy to grasp for many people, so do not get bent out of shape over a problem. Thanks Helpful 100 Not Helpful 25
• If an instructor tells you to draw a free body diagram, be sure that that is exactly what you draw. Thanks Helpful 89 Not Helpful 24

Things You'll Need

• A Writing Utensil (preferably a pencil or erasable pen of sorts)
• Calculator with all the functions you need for your exam
• An understanding of the equations needed to solve the problems. Or a list of them will suffice if you are just trying to get through the course alive.

You Might Also Like

Expert Interview

Thanks for reading our article! If you’d like to learn more about teaching, check out our in-depth interview with Sean Alexander, MS .

• ↑ https://iopscience.iop.org/article/10.1088/1361-6404/aa9038
• ↑ https://physics.wvu.edu/files/d/ce78505d-1426-4d68-8bb2-128d8aac6b1b/expertapproachtosolvingphysicsproblems.pdf
• ↑ https://www.brighthubeducation.com/science-homework-help/42596-tips-to-choosing-the-correct-physics-formula/

About This Article

• Send fan mail to authors

Reader Success Stories

Monish Shetty

Sep 30, 2016

Did this article help you?

Dec 6, 2023

Oct 9, 2017

Komal Verma

Dec 6, 2017

Lowella Tabbert

Jun 7, 2022

Featured Articles

Trending Articles

Watch Articles

• Terms of Use
• Privacy Policy
• Do Not Sell or Share My Info
• Not Selling Info

wikiHow Tech Help Pro:

Level up your tech skills and stay ahead of the curve

PHYSICS PROBLEM SOLVING STRATEGY Dr. Mark Hollabaugh Normandale Community College

http://www.nr.cc.mn.us/physics/Faculty/HOLLABGH/probsolv.htm

    Two factors can help make you a better physics problem solver.  First of all, you must know and understand the principles of physics.  Secondly, you must have a strategy for applying these principles to new situations in which physics can be helpful.  We call these situations problems.  Many students say, “I understand the material, I just can’t do the problems.”  If this is true of you as a physics student, then maybe you need to develop your problem-solving skills.  Having a strategy to organize those skills can help you.

    Physics problem solving can be learned just like you learned to drive a car, play a musical instrument, or ride a bike.  What can aid you more than anything is to have a general approach to follow with each problem you encounter.  You may use different tools or tactics with differing areas of physics, but the overall strategy remains the same.  Most likely, you have already acquired some problem-solving skills and habits from previous courses in physics, chemistry, or mathematics.  Like other areas of learning and life, some of these habits may be beneficial and some may actually hinder your progress in learning how to solve physics problems.

    So, in learning this new approach, be willing to try new ideas and to discard old habits that may in fact be hindering your understanding.  As you mature as a physics problem solver, you will find that the approach will become second nature to you.  You will begin automatically to do those things that will lead you to construct an effective solution to the problem.

    As with so many other learning activities, it is useful to break a problem solving strategy into major and minor steps.  The strategy we would like you to learn has five major steps:  Focus the Problem , Physics Description , Plan a Solution , Execute the Plan , and Evaluate the Solution .  Let’s take a detailed look at each of these steps and then do an sample problem following the strategy.  At this stage of our discussion, do not worry if there are physics terms or concepts that you do not understand.  You will learn these concepts as they are needed.  Then, refer back to this discussion.

FOCUS the PROBLEM      Usually when you read the statement of a physics problem, you must visualize the objects involved and their context.  You need to draw a picture and indicate any given information.

(1) First, construct a mental image of the problem situation. (2)Next draw a rough, although literal, picture showing the important objects, their motion, and their interactions.  An interaction, for example, may consist of one object being connected to another by a rope. (3) Label all known information.  At this point, do not worry about assigning algebraic symbols to specific quantities.

    Sometimes the question being asked in the problem is not obvious.  “Is the rope safe?” is not something you can directly answer.  Ask yourself, what specifically is being asked?  How does this translate into some calculable quantity?

    There are many ways to solve a physics problem.  One part of learning how to solve a problem is to know what approach to use.  You will need to outline the concepts and principles you think will be useful in solving the problem.

If simple motions are involved, use the kinematics definition of velocity and acceleration. If forces are involved and objects interact due to these forces, use Newton's Laws of Motion. Forces that act over a time interval and cause objects to change their velocities suggests using the Conservation of Momentum. Frequently in situations involving thermal physics or electromagnetism, the principle of Conservation of Energy is useful. You may need to specify time intervals over which the application of each principle will be the most useful. It is important to identify any constraints present in this situation, such as “the car doesn’t leave the road”. Specify any approximations or simplifications you think will make the problem solution easier, but will not affect the result significantly.  Frequently we ignore frictional forces due to air resistance.

    Your approach probably will be very consistent throughout a particular section of the textbook.  The challenge for you will be to apply the approach in a variety of situations.

DESCRIBE the PHYSICS      A “physics description” of a problem translates  the given information and a very literal picture into an idealized diagram and defines variables that can be manipulated to calculate desired quantities.  In a sense, you are translating the literal situation into an idealized situation where you can then apply the laws the physics.  The biggest shortcoming of beginning physics problem solvers is attempting to apply the laws of physics, that is write down equations, before undertaking this qualitative analysis of the problem.  If you can resist the temptation to look for equations too early in your problem solution, you will become a much more effective problem solver.

To construct your physics description, you must do the following:

• Translate your picture into a diagram(s) which gives only the essential information for a mathematical solution.  In an idealized diagram, people, cars, and other objects may become square blocks or points.
• Define a symbol for every important physics variable on your diagram.
• Usually you need to draw a coordinate system showing the + and - directions.
• If you are using kinematics concepts, draw a motion diagram specifying the objects’ velocity and acceleration at definite positions and times.
• If interactions are important, draw idealized, free body, and force diagrams.
• When using conservation principles, draw "before", "transfer" (i.e., during), and "after" diagrams to show how the system changes. To the side of your diagram(s), give the value for each physics variable you have labeled on the diagram(s) or specify that it is unknown.

    Then, using the question, your physics description and the approach you have stated, you will need to identify a target variable.  That is, you must decide what unknown quantity is it that you must calculate from your list of variables.  Ask yourself if the calculated quantity answers the question.  In complex problems there may be more than one target variable or some intermediate variables you will calculate.

     Now, knowing the target variable(s), and your approach, you can assemble your toolbox of mathematical expressions using the principles and constraints from your approach to relate the physics variables from your diagrams.  This is the first time you really begin to look for quantitative relationships among the variables.

 PLAN the SOLUTION     Before you actually begin to calculate an answer, take time to make a plan.  Usually when the laws of physics are expressed in an equation, the equation is a general, universal statement.  You must construct specific algebraic equations that will enable you to calculate the target variable.

• Determine how the equations in your toolbox can be combined to find your target variable. Begin with an equation containing the target variable.
• Identify any unknowns in that equation.
• Find equations from your toolbox containing these unknowns.
• Continue this process until your equations contain no new unknowns.
• Number each equation for easy reference.
• Do not solve equations numerically at this time.

    Frequently expert problem solvers will start with the target variable and work backwards to determine a path to the answer.  Sometimes the units will help you find the correct path.  For example, if you are looking for a velocity, you know your final answer must be in m/s.

    You have a solution if your plan has as many independent equations as there are unknowns.  If not, determine other equations or check the plan to see if it is likely that a variable will cancel from your equations.

    If you have the same number of equations and unknowns, indicate the order in which to solve the equations algebraically for the target variable. Typically, you begin your construction of the plan at the end and work backwards to the first step,  That is, you write down the equation containing the target variable first.

EXECUTE the PLAN     Now you are ready to execute the plan.

• Do the algebra in the order given by your outline.
• When you are done you should have a single equation with your target variable isolated on one side and only known quantities on the other side.
• Substitute the values (numbers with units) into this final equation.
• Make sure units are consistent so that they will cancel properly.

    Finally, calculate the numerical result for the target variable(s).  Make sure your final answer is clear to the person who will evaluate your solution.

    It is extremely important to solve the problem algebraically before inserting any numerical values.  Some unknown quantities may cancel out and you won’t need to actually know their numerical value.  In some complex problems it can be useful to calculate intermediate numerical results as a check on the reasonableness of your solution.

EVALUATE the SOLUTION      Finally, you are ready to evaluate your answer.  Here, you must use your common sense about how the real world works as well as those aspects of the physical world you have learned in your physics class.

• Do vector quantities have both magnitude and direction?
• Can someone else follow your solution?
• Is the result reasonable and within your experience?  Remember, for example, that cars don’t travel down the highway at 300 mi/hr.  If you put a cooler object into hot water, the water cools down and the object rises in temperature.
• Do the units make sense?  Velocity is not measured, for example, in kg/s.
• Have you answered the question?

    Whenever possible, it is a good idea to read through the solution carefully, especially if it is being evaluated by your instructor.  If your evaluation suggests to you that your answer is incorrect or unreasonable, make a statement to that effect and explain your reasoning.  

Further Reading:

Patricia Heller, Ronald Keith, and Scott Anderson (1992), Teaching Problem Solving Through Cooperative Grouping.  Part 1:  Group Versus Individual Problem Solving, American Journal of Physics , Vol. 60, No. 7, pp. 627-636.

Patricia Heller and Mark Hollabaugh (1992), Teaching Problem Solving Through Cooperative Grouping.  Part 2:  Designing Problems and Structuring Groups,  American Journal of Physics , Vol. 60, No. 7, pp. 637-644.  

share this!

June 24, 2024

This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility:

fact-checked

trusted source

New mathematical proof helps to solve equations with random components

by Kathrin Kottke, University of Münster

Whether it's physical phenomena, share prices or climate models—many dynamic processes in our world can be described mathematically with the aid of partial differential equations. Thanks to stochastics—an area of mathematics which deals with probabilities—this is even possible when randomness plays a role in these processes.

Something researchers have been working on for some decades now are so-called stochastic partial differential equations. Working together with other researchers, Dr. Markus Tempelmayr at the Cluster of Excellence Mathematics Münster at the University of Münster has found a method which helps to solve a certain class of such equations.

The results have been published in the journal Inventiones mathematicae .

The basis for their work is a theory by Prof. Martin Hairer, recipient of the Fields Medal, developed in 2014 with international colleagues. It is seen as a great breakthrough in the research field of singular stochastic partial differential equations. "Up to then," Tempelmayr explains, "it was something of a mystery how to solve these equations. The new theory has provided a complete 'toolbox,' so to speak, on how such equations can be tackled."

The problem, Tempelmayr continues, is that the theory is relatively complex, with the result that applying the 'toolbox' and adapting it to other situations is sometimes difficult.

"So, in our work, we looked at aspects of the 'toolbox' from a different perspective and found and proved a method which can be used more easily and flexibly."

The study, in which Tempelmayr was involved as a doctoral student under Prof. Felix Otto at the Max Planck Institute for Mathematics in the Sciences, published in 2021 as a pre-print. Since then, several research groups have successfully applied this alternative approach in their research work.

Stochastic partial differential equations can be used to model a wide range of dynamic processes, for example, the surface growth of bacteria, the evolution of thin liquid films, or interacting particle models in magnetism. However, these concrete areas of application play no role in basic research in mathematics as, irrespective of them, it is always the same class of equations which is involved.

The mathematicians are concentrating on solving the equations in spite of the stochastic terms and the resulting challenges such as overlapping frequencies which lead to resonances.

Various techniques are used for this purpose. In Hairer's theory, methods are used which result in illustrative tree diagrams. "Here, tools are applied from the fields of stochastic analysis, algebra and combinatorics," explains Tempelmayr. He and his colleagues selected, rather, an analytical approach . What interests them in particular is the question of how the solution of the equation changes if the underlying stochastic process is changed slightly.

The approach they took was not to tackle the solution of complicated stochastic partial differential equations directly, but, instead, to solve many different simpler equations and prove certain statements about them.

"The solutions of the simple equations can then be combined—simply added up, so to speak—to arrive at a solution for the complicated equation which we're actually interested in." This knowledge is something which is used by other research groups who themselves work with other methods.

Provided by University of Münster

Explore further

Feedback to editors

Early childhood problems linked to persistent school absenteeism

8 hours ago

Researchers find genetic stability in a long-term Panamanian hybrid zone of manakins

10 hours ago

Detective work enables Perseverance Mars rover team to revive SHERLOC instrument

NASA's Juno probe gets a close-up look at lava lakes on Jupiter's moon Io

Simple new process stores carbon dioxide in concrete without compromising strength

Surprising phosphate finding in NASA's OSIRIS-REx asteroid sample

11 hours ago

First case of Down syndrome in Neanderthals documented in new study

Understanding quantum states: New research shows importance of precise topography in solid neon qubits

New study reveals comet airburst evidence from 12,800 years ago

12 hours ago

Time-compression in electron microscopy: Terahertz light controls and characterizes electrons in space and time

13 hours ago

Relevant PhysicsForums posts

Views on complex numbers.

3 hours ago

P-adic numbers and the Ramanujan summation

20 hours ago

Is PI (##\pi##) really a number?

Jun 25, 2024

Aspects Behind the Concept of Dimension in Various Fields

Jun 23, 2024

Why are the axes taken as perpendicular to each other?

Jun 20, 2024

Memorizing trigonometric identities

May 26, 2024

More from General Math

Related Stories

Mathematician discovers method to simplify polymer growth modelling

Nov 12, 2019

A mathematical bridge between the huge and the tiny

Apr 29, 2024

Mathematician proposes a new criterion for solving the Boussinesq equations

Jan 24, 2020

Mathematician suggests a scheme for solving telegraph equations

Feb 11, 2021

Mathematicians proposed an express method for calculation of the propagation of light

Sep 13, 2019

Mathematicians create a method for studying the properties of porous materials

Jan 30, 2020

Recommended for you

Merging AI and human efforts to tackle complex mathematical problems

Jun 24, 2024

Study finds cooperation can still evolve even with limited payoff memory

Jun 19, 2024

Study shows the power of social connections to predict hit songs

Jun 11, 2024

Wire-cut forensic examinations currently too unreliable for court, new study says

Jun 10, 2024

How can we make good decisions by observing others? A videogame and computational model have the answer

Jun 4, 2024

Data scientists aim to improve humanitarian support for displaced populations

Jun 3, 2024

Let us know if there is a problem with our content

Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ).

Please select the most appropriate category to facilitate processing of your request

Thank you for taking time to provide your feedback to the editors.

Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages.

E-mail the story

Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form.

Newsletter sign up

Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties.

More information Privacy policy

Donate and enjoy an ad-free experience

We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account.

E-mail newsletter

Help | Advanced Search

Computer Science > Machine Learning

Title: kolmogorov arnold informed neural network: a physics-informed deep learning framework for solving pdes based on kolmogorov arnold networks.

Abstract: AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN). We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP in terms of accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

Submission history

Access paper:.

• HTML (experimental)
• Other Formats

References & Citations

• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

latest in US News

'Fake' roadside memorial set up with busted kid's bike, photo to...

Woman involved in Cleveland Guardians stadium fight sets record...

NYPD Chief of Patrol cleared by department's watchdog panel over...

Ashley Biden mocked after struggling with LGBTQ acronym at White...

Texas mom claims she was kicked off United plane for accidentally...

Trump holds 4-point lead over Biden ahead of first debate — his...

The top 10 places to go off-the-grid in the US — does your...

Driver walks away unscathed after plane crashes into her pickup...

Budding lawmaker smirks after allegedly tossing tarantula at accused squatter: ‘creatively solving problems’.

• View Author Archive
• Get author RSS feed

Thanks for contacting us. We've received your submission.

A budding Minnesota lawmaker was pictured smirking in her mugshot after she was arrested for allegedly tossing a live tarantula at her renter during a fight.

Marisa Simonetti, 30, was cuffed on an assault charge after she was accused of hurling the eight-legged critter inside the home in Edina, just outside Minneapolis, last Friday, FOX9 reported.

The web-tangling crime allegedly unfolded when the housemate, local attorney Jackie Vasquez, called 911 to report that Simonetti was trespassing in her section of the property.

When cops arrived, the alleged victim told them Simonetti — who is running as a Hennepin County Board candidate — had thrown the spider and other pieces of junk at her during a rental dispute.

Footage posted on social media showed Simonetti dumping the tarantula out of a container on a set of stairs inside the home.

Simonetti, for her part, has claimed the fight erupted when she accused Vasquez of being a squatter.

The wannabe lawmaker, who doesn’t own the property, said she rented a room to Vasquez several weeks earlier via a short-term rental website — but she refused to get out when the contract was up.

“Perhaps I should have invited her up for tea and crumpets,” Simonetti told the outlet about the fight.

Simonetti spent the weekend in custody and was cut loose following a court hearing Monday.

Despite the spider saga, Simonetti said she still has every intention of running for Hennepin County commissioner.

“I’m good at creatively solving problems, and at the end of the day, I didn’t physically harm anybody,” she said.

“I’m a little unconventional in my ways — sometimes. I mean, I’m a silly goose.”

Share this article:

Advertisement

• Advanced Search

Helmholtz decomposition based windowed Green function methods for elastic scattering problems on a half-space

New citation alert added.

This alert has been successfully added and will be sent to:

You will be notified whenever a record that you have chosen has been cited.

To manage your alert preferences, click on the button below.

New Citation Alert!

Please log in to your account

Information & Contributors

Bibliometrics & citations, view options, recommendations, nyström method for elastic wave scattering by three-dimensional obstacles.

Nyström method is developed to solve for boundary integral equations (BIE's) for elastic wave scattering by three-dimensional obstacles. To generate the matrix equation from a BIE, Nyström method applies a quadrature rule to the integrations of smooth ...

Multilevel fast multipole algorithm for elastic wave scattering by large three-dimensional objects

Multilevel fast multipole algorithm (MLFMA) is developed for solving elastic wave scattering by large three-dimensional (3D) objects. Since the governing set of boundary integral equations (BIE) for the problem includes both compressional and shear ...

Numerical Solution of the Helmholtz Equation in 2D and 3D Using a High-Order Nyström Discretization

We show how to solve time-harmonic scattering problems by means of a high-order Nystr m discretization of the boundary integral equations of wave scattering in 2D and 3D. The novel aspect of our new method is its use of local corrections to the ...

Information

Published in.

Academic Press Professional, Inc.

United States

Publication History

Author tags.

• Elastic scattering
• Windowed Green function
• Boundary integral equation
• Research-article

Contributors

Other metrics, bibliometrics, article metrics.

• 0 Total Citations
• 0 Total Downloads
• Downloads (Last 12 months) 0
• Downloads (Last 6 weeks) 0

View options

Login options.

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Share this publication link.

Copying failed.

Share on social media

Affiliations, export citations.

• Please download or close your previous search result export first before starting a new bulk export. Preview is not available. By clicking download, a status dialog will open to start the export process. The process may take a few minutes but once it finishes a file will be downloadable from your browser. You may continue to browse the DL while the export process is in progress. Download
• Download citation
• Copy citation

We are preparing your search results for download ...

We will inform you here when the file is ready.

Your file of search results citations is now ready.

Your search export query has expired. Please try again.