This page written by Dan Styer, Oberlin CollegePhysics Department;
http://www.oberlin.edu/physics/dstyer/SolvingProblems.html;
last updated 22 April 1999.
The problems and examinations in this physics course exercise notonly your knowledge of physics but also your skill in solvingproblems. Professional physicists earn their salaries notparticularly for their knowledge of physics but for their ability tosolve workplace problems. This document presents tips for honing yourproblem solving skills. These tips and techniques will prove usefulto you in your physics courses, in your other college courses, inyour career, and in your everyday life.
To set the stage, I want to discuss an example of problem solvingfrom everyday life, namely building a jigsaw puzzle. There are anumber of different approaches to building a jigsaw puzzle: Myapproach is to first turn all the pieces face up, then put togetherthe edge pieces to make a frame, then sort the remaining pieces intopiles corresponding to small "sub-puzzles" (blue pieces over here,red pieces over there). I build the sub-puzzles, then piece thesub-puzzles together to build the whole thing. Other people havedifferent approaches to building jigsaw puzzles, but nobody,nobody, builds a puzzle by picking up the first piece andputting it in exactly the correct position, then picking up thesecond piece and putting it in exactly the correct position, and soforth. Solving a jigsaw puzzle involves an approach--a strategy--anda lot of "creative fumbling" as well.
Your physics textbook contains many solved "sample problems". Thesolutions presented there are analogous to the completed jigsawpuzzle, with every piece in its proper position. No one solves aphysics problem by simply writing down the correct equations and thecorrect reasoning with the correct connections the first timethrough, just as no one builds a jigsaw puzzle by putting every piecein its correct position the first time through. The "solved problems"in your book are extraordinarily valuable and they deserve yourcareful study, but they represent the end product of a problemsolving session and they rarely show the process involved in reachingthat end product. This document aims to expose you to theprocess.
Solving a physics problem usually breaks down into threestages:
This document treats each of these three elements in turn, andconcludes with a summary.
Look before you leap. Whenever you face a problem, there isan immediate temptation to rush in, roll up your sleeves, and begintinkering with it. Resist that temptation. If you start yourdetailed work--the execution stage--immediately, you will likelywrite down a lot of correct statements that do not lead to an answer.Instead, think about the problem on an overview level. Whatsort of conceptual tools will you need to solve the problem? Whatpath will you take to the solution, and in what direction should youstart off? Concretely, it often helps to classify your problem byits method of solution.
If you are looking for a child lost in the woods, your first stepis to sit down, think about what the child probably did and where heprobably is, and devise a strategy that will allow you to effectivelyrescue him. If, instead, you just rush about the woods in randomdirections, you're likely to become lost yourself.
Where are you now, and where do you want to go? Before youcan design a path that takes you from the statement of the problem toits answer, you must be clear about what the situation is and whatthe goals are. It often helps to check off each given datum ofthe problem, and to underline the objective. But for gettingan overall sense of the problem, nothing beats summarizing the wholesituation with a diagram. The diagram will organize your work andsuggest ways to proceed. One of my course graders told me that "Whenstudents draw a diagram and label it carefully, they are forced tothink about what's going on, and they usually do well. If they justtry a globule of math, they mess up."
Keep the goal in sight. Don't get caught in blind alleysthat lead nowhere, or even in broad boulevards that lead somewherebut not to where you want to go. It sometimes helps to map a strategybackwards, by saying: "I want to find the answer Z. If I knewY I could find Z. If I knew X I could findY . . . " and so forth until you get back to something you aregiven in the problem statement.
Some students find it useful to make a list of the informationgiven and the goal to be uncovered (e.g. "given the constantacceleration, the initial velocity, and the time, find thedisplacement"). Others find it sufficient to write down only the goal(e.g. "to find: displacement").
Ineffective strategy. Do not page through your book lookingfor a magic formula that will give you the answer. Physics teachersdo not assign problems in order to torture innocent young minds . . .they assign problems in order to force you into active, intimateinvolvement with the concepts and tools of physics. Rarely is suchinvolvement provided by plugging numbers into a single equation,hence rarely will you be assigned a problem that yields to thisattack. In those rare instances when you do face a problem that canbe solved by plugging numbers into a formula, the most effective wayto find that formula is by thinking about the physical principlesinvolved, not by flipping through the pages in your book.
Make the problem more specific. You're asked to find thenumber of ways that M balls can be placed into Nbuckets. Suppose you can't even begin to map out a strategy. Then trythe problem of 3 balls in 5 buckets. Solving the more specificproblem will give you clues on how to solve the more general problem.And once you use those clues to solve the more general problem, youcan check your solution by trying it out for the already-solvedspecial case M=3 and N=5.
Large problems. At times you will be faced with bigproblems for which no method of solution is immediately apparent. Inthis case, break your problem into several smaller subproblems, eachof which is simple enough that you know how to solve it. At thisstrategy-design stage it is not important that you actually solve thesubproblems, but rather that you know you can solve them. You mightbegin by mapping out a strategy that leads nowhere, but then youhaven't wasted time by implementing this strategy. Once you havemapped out a strategy that leads from the given information to theanswer, you can then go back and execute the calculations. Thisstrategy has been known from the time of the ancients under the nameof "divide and conquer".
Eventually, of course, you do have to roll up your sleevesand tinker with the problem. As you do so, keep your strategy inmind, and keep the following tips in mind as well:
Work with symbols. Depending on the problem statement, thefinal answer might be a formula or a number. In either case, however,it's usually easier to work the problem with symbols and plug innumbers, if requested, only at the very end. There are three reasonsfor this: First, it's easier to perform algebraic manipulations on asymbol like "m" than on a value like "2.59 kg". Second, itoften happens that intermediate quantities cancel out in the finalresult. Most important, expressing the result as an equation enablesyou to examine and understand it (see the section on "AnswerChecking") in a way that a number alone does not permit.
(Working with symbols instead of numbers can lead to confusion asto which symbols represent given information and which representunknown desired answers. You can resolve this difficulty byremembering--as recommended above--to "keep the goal in sight".)
Define symbols with mnemonic names. If a problem involves ahelium atom colliding with a gold atom, then definemh as the mass of the helium atom andmg as the mass of the gold atom. If you insteadpick the symbols m1 and m2, youstand a good chance of mixing up the symbols and their meanings asyou solve the problem. And if you don't define the symbols at all,but just begin throwing around m's and M's, you'llconfuse both yourself and whoever is grading your answer.
Keep packets of related variables together. In accelerationproblems, the quantity (1/2)at2 comes up over andover again. This collection of variables has a simple physicalinterpretation, transparent dimensions, and a convenient memorableform. In short, it is easy to work with as a packet. Take advantageof this ease. Don't artificially divide this packet into pieces, orwrite it in an unfamiliar form like t2a/2.Packets like this come up in all aspects of physics--some are evengiven names (e.g. "the Bohr radius" in atomic physics). Look forthese packets, think about what they are telling you, and respecttheir integrity.
Neatness and organization. I am not your mother, and I willnot tell you how to organize either your dorm room or your problemsolutions. But I can tell you that it is easier to work from neat,well-organized pages than from scribbles. I can also warn you aboutcertain handwriting pitfalls: Distinguish carefully between tand +, between l and 1, and between Z and 2. (Iwrite a t with a loop at the bottom, an l in scriptlettering, and a Z with a cross bar. You can form your ownconventions.) These suggestions on neatness, organization, andhandwriting do not arise from prudishness--they are practicalsuggestions that help avoid algebraic errors, and they are for yourbenefit, not mine. (On the other hand, it doesn't hurt to be neat andorganized for the benefit of your grader. One course grader of minepointed out: "If I can't read it, I can't give you credit.")
Avoid needless conversions. If the problem gives you onelength in meters and another in inches, then it's probably best toconvert all lengths to meters. But if all the lengths are in inches,then there's no need to convert everything to meters--your answershould be in inches. In fact, you might not actually need to convert.For example, perhaps two lengths are given in inches and the finalanswer turns out to depend only on the ratio of those two lengths. Inthat case, the ratio is the same whether the lengths going into theratio are inches or meters. It's easy to make arithmetic errors whiledoing conversions. If you don't convert, then you don't make thoseerrors!
Keep it simple. I will not assign baroque problems thatrequire tortuous explanations and pages of algebra. If you findyourself working in such a way, then you're on the wrong path. Thecure is to stop, go back to the beginning, and start over with a newstrategy.
Checking your answer does not mean comparing it to the answer inthe back of the book. It means finding the characteristics of youranswer and comparing them to the characteristics that you expect.Some of your problems--particularly the ones assigned early in thecourse--will actually lead you through the checking stage in order tofamiliarize you with the process. Other problems will leave it to youto perform this check. In either case, checking your answer is notjust good problem solving practice that helps you gain points onproblem assignments and on exams. The checking stage buildsfamiliarity with the content of physics and the character of problemsolutions, and hence develops your intuition to make solving otherproblems--and learning more physics--easier. (See Daniel F. Styer,"Guest comment: Getting there is half the fun", American Journalof Physics 64 (1998) 105-106.)
Dimensional analysis. Suppose you find a formula fordistance (in, say, meters) in terms of some information aboutvelocity (meters/second), acceleration (meters/second2),and time (seconds). If your formula is correct then all of thedimensions on the right hand side must cancel so as to end up with"meters".
Numerical reasonableness. If your problem asks you to findthe mass of a squirrel, do you find a mass of 1,970 kilograms? Evenworse, do you find a mass of -1,970 kilograms?
Algebraically possible. Would evaluating your formula everlead you to divide by zero or take the square root of negativenumber?
Functionally reasonable. Does your answer depend on thegiven quantities in a reasonable way? For example, you might be askedhow far a projectile travels after it is launched at a given speedwith a given angle. Common sense says that if the initial speed isincreased (keeping the angle constant) then the distance traveledwill increase. Does your formula agree with common sense?
Limiting values and special cases. In the projectile traveldistance problem mentioned above, the range is obviously zero for avertical launch. Does your formula give this result? If you solve aproblem regarding two objects, does it give the proper result whenthe two objects have equal masses? When one of them has zero mass(i.e. does not exist)?
Symmetry. Problems often have geometrical symmetry fromwhich you can determine the direction of a vector but not itsmagnitude. More often they have a "permutation" symmetry: If yourproblem has two objects, you can call the cube "object number 1" andthe sphere "object number 2" but your final answer had better notdepend upon how you numbered your objects. (That is, it should givethe same answer if every "1" is changed to a "2" and vice versa.)
Specify units. "The distance is 5.72" is not an answer. Isthat 5.72 miles, 5.72 meters, or 5.72 inches? Similarly, if theanswer is a vector, both magnitude and direction must be specified.(The direction may be drawn into a diagram rather than statedexplicitly.)
Significant figures. Any number that comes from anexperiment comes with some uncertainty. Most of the numbers in thiscourse come with three significant figures. If a ball rolls 3.42meters in 2.41 seconds, then report its speed as 1.34 m/s, not1.34439834 m/s. Most introductory physics courses do not require aformal or technical error analysis, but you should avoid inaccuratestatements like the second quotient above.
Large problems. If you break up your large problem intoseveral subproblems, as recommended above, then check your results atthe end of each subproblem. If your answer to the second subproblempasses its checks, but your answer to the third subproblem fails itschecks, then your execution error almost certainly falls within thethird subproblem. Knowing its general location, you can quickly goback and correct the error, so its effects will not propagate on tothe remaining subproblems. This can be a real time-saver.
The problems in your physics course can be fun and exciting.Approach them in the spirit of exploration and they will notdisappoint you!
The classic exploration of mathematical problem solving techniqueis
More mundane and somewhat pedantic, but nevertheless valuable,is
Study of the following books will help develop your general (asopposed to strictly mathematical) problem-solving skills:
Entry into recent literature on physics problem solving skills isprovided by