The problems and examinations in this physics course exercise not only your
knowledge of physics but also your skill in solving problems.
Professional physicists earn their salaries not particularly for
their knowledge of physics but for their ability to solve
workplace problems. This document presents tips for honing your
problem solving skills. These tips and techniques will prove
useful to you in your physics courses, in your other
college courses, in your career, and
in your everyday life.
To set the stage, I want to discuss an example of problem solving from everyday life, namely building a jigsaw puzzle. There are a number of different approaches to building a jigsaw puzzle: My approach is to first turn all the pieces face up, then put together the edge pieces to make a frame, then sort the remaining pieces into piles corresponding to small "sub-puzzles" (blue pieces over here, red pieces over there). I build the sub-puzzles, then piece the sub-puzzles together to build the whole thing. Other people have different approaches to building jigsaw puzzles, but nobody, nobody, builds a puzzle by picking up the first piece and putting it in exactly the correct position, then picking up the second piece and putting it in exactly the correct position, and so forth. Solving a jigsaw puzzle involves an approach--a strategy--and a lot of "creative fumbling" as well.
Your physics textbook contains many solved "sample problems". The solutions presented there are analogous to the completed jigsaw puzzle, with every piece in its proper position. No one solves a physics problem by simply writing down the correct equations and the correct reasoning with the correct connections the first time through, just as no one builds a jigsaw puzzle by putting every piece in its correct position the first time through. The "solved problems" in your book are extraordinarily valuable and they deserve your careful study, but they represent the end product of a problem solving session and they rarely show the process involved in reaching that end product. This document aims to expose you to the process.
Solving a physics problem usually breaks down into three stages:
1. Design a strategy.
2. Execute that strategy.
3. Check the resulting answer.
This document treats each of these three elements in turn, and concludes with a summary.
Strategy Design: Look before you leap. Whenever you face a problem, there is an immediate temptation to rush in, roll up your sleeves, and begin tinkering with it. Resist that temptation. If you start your detailed work--the execution stage--immediately, you will likely write down a lot of correct statements that do not lead to an answer. Instead, think about the problem on an overview level. What sort of conceptual tools will you need to solve the problem? What path will you take to the solution, and in what direction should you start off? Concretely, it often helps to classify your problem by its method of solution.
If you are looking for a child lost in the woods, your first step is to sit down, think about what the child probably did and where he probably is, and devise a strategy that will allow you to effectively rescue him. If, instead, you just rush about the woods in random directions, you're likely to become lost yourself.
Where are you now, and where do you want to go? Before you can design a path that takes you from the statement of the problem to its answer, you must be clear about what the situation is and what the goals are. It often helps to check off each given datum of the problem, and to underline the objective. But for getting an overall sense of the problem, nothing beats summarizing the whole situation with a diagram. The diagram will organize your work and suggest ways to proceed. One of my course graders told me that "When students draw a diagram and label it carefully, they are forced to think about what's going on, and they usually do well. If they just try a globule of math, they mess up."
Keep the goal in sight. Don't get caught in blind alleys that lead nowhere, or even in broad boulevards that lead somewhere but not to where you want to go. It sometimes helps to map a strategy backwards, by saying: "I want to find the answer Z. If I knew Y I could find Z. If I knew X I could find Y . . . " and so forth until you get back to something you are given in the problem statement.
Some students find it useful to make a list of the information given and the goal to be uncovered (e.g. "given the constant acceleration, the initial velocity, and the time, find the displacement"). Others find it sufficient to write down only the goal (e.g. "to find: displacement").
Ineffective strategy. Do not page through your book looking for a magic formula that will give you the answer. Physics teachers do not assign problems in order to torture innocent young minds . . . they assign problems in order to force you into active, intimate involvement with the concepts and tools of physics. Rarely is such involvement provided by plugging numbers into a single equation, hence rarely will you be assigned a problem that yields to this attack. In those rare instances when you do face a problem that can be solved by plugging numbers into a formula, the most effective way to find that formula is by thinking about the physical principles involved, not by flipping through the pages in your book.
Make the problem more specific. You're asked to find the number of ways that M balls can be placed into N buckets. Suppose you can't even begin to map out a strategy. Then try the problem of 3 balls in 5 buckets. Solving the more specific problem will give you clues on how to solve the more general problem. And once you use those clues to solve the more general problem, you can check your solution by trying it out for the already-solved special case M=3 and N=5.
Large problems. At times you will be faced with big problems for which no method of solution is immediately apparent. In this case, break your problem into several smaller sub-problems, each of which is simple enough that you know how to solve it. At this strategy-design stage it is not important that you actually solve the sub-problems, but rather that you know you can solve them. You might begin by mapping out a strategy that leads nowhere, but then you haven't wasted time by implementing this strategy. Once you have mapped out a strategy that leads from the given information to the answer, you can then go back and execute the calculations. This strategy has been known from the time of the ancients under the name of "divide and conquer".
Execution (Tactics): Eventually, of course, you do have to roll up your sleeves and tinker with the problem. As you do so, keep your strategy in mind, and keep the following tips in mind as well:
Work with symbols. Depending on the problem statement, the final answer might be a formula or a number. In either case, however, it's usually easier to work the problem with symbols and plug in numbers, if requested, only at the very end. There are three reasons for this: First, it's easier to perform algebraic manipulations on a symbol like "m" than on a value like "2.59 kg". Second, it often happens that intermediate quantities cancel out in the final result. Most important, expressing the result as an equation enables you to examine and understand it (see the section on "Answer Checking") in a way that a number alone does not permit.
(Working with symbols instead of numbers can lead to confusion as to which symbols represent given information and which represent unknown desired answers. You can resolve this difficulty by remembering--as recommended above--to "keep the goal in sight".)
Define symbols with mnemonic names. If a problem involves a helium atom colliding with a gold atom, then define mh as the mass of the helium atom and mg as the mass of the gold atom. If you instead pick the symbols m1 and m2, you stand a good chance of mixing up the symbols and their meanings as you solve the problem. And if you don't define the symbols at all, but just begin throwing around m's and M's, you'll confuse both yourself and whoever is grading your answer.
Keep packets of related variables together. In acceleration problems, the quantity (1/2)at2 comes up over and over again. This collection of variables has a simple physical interpretation, transparent dimensions, and a convenient memorable form. In short, it is easy to work with as a packet. Take advantage of this ease. Don't artificially divide this packet into pieces, or write it in an unfamiliar form like t2a/2. Packets like this come up in all aspects of physics--some are even given names (e.g. "the Bohr radius" in atomic physics). Look for these packets, think about what they are telling you, and respect their integrity.
Image Credit: martinofranchi
To set the stage, I want to discuss an example of problem solving from everyday life, namely building a jigsaw puzzle. There are a number of different approaches to building a jigsaw puzzle: My approach is to first turn all the pieces face up, then put together the edge pieces to make a frame, then sort the remaining pieces into piles corresponding to small "sub-puzzles" (blue pieces over here, red pieces over there). I build the sub-puzzles, then piece the sub-puzzles together to build the whole thing. Other people have different approaches to building jigsaw puzzles, but nobody, nobody, builds a puzzle by picking up the first piece and putting it in exactly the correct position, then picking up the second piece and putting it in exactly the correct position, and so forth. Solving a jigsaw puzzle involves an approach--a strategy--and a lot of "creative fumbling" as well.
Your physics textbook contains many solved "sample problems". The solutions presented there are analogous to the completed jigsaw puzzle, with every piece in its proper position. No one solves a physics problem by simply writing down the correct equations and the correct reasoning with the correct connections the first time through, just as no one builds a jigsaw puzzle by putting every piece in its correct position the first time through. The "solved problems" in your book are extraordinarily valuable and they deserve your careful study, but they represent the end product of a problem solving session and they rarely show the process involved in reaching that end product. This document aims to expose you to the process.
Solving a physics problem usually breaks down into three stages:
1. Design a strategy.
2. Execute that strategy.
3. Check the resulting answer.
This document treats each of these three elements in turn, and concludes with a summary.
Strategy Design: Look before you leap. Whenever you face a problem, there is an immediate temptation to rush in, roll up your sleeves, and begin tinkering with it. Resist that temptation. If you start your detailed work--the execution stage--immediately, you will likely write down a lot of correct statements that do not lead to an answer. Instead, think about the problem on an overview level. What sort of conceptual tools will you need to solve the problem? What path will you take to the solution, and in what direction should you start off? Concretely, it often helps to classify your problem by its method of solution.
If you are looking for a child lost in the woods, your first step is to sit down, think about what the child probably did and where he probably is, and devise a strategy that will allow you to effectively rescue him. If, instead, you just rush about the woods in random directions, you're likely to become lost yourself.
Where are you now, and where do you want to go? Before you can design a path that takes you from the statement of the problem to its answer, you must be clear about what the situation is and what the goals are. It often helps to check off each given datum of the problem, and to underline the objective. But for getting an overall sense of the problem, nothing beats summarizing the whole situation with a diagram. The diagram will organize your work and suggest ways to proceed. One of my course graders told me that "When students draw a diagram and label it carefully, they are forced to think about what's going on, and they usually do well. If they just try a globule of math, they mess up."
Keep the goal in sight. Don't get caught in blind alleys that lead nowhere, or even in broad boulevards that lead somewhere but not to where you want to go. It sometimes helps to map a strategy backwards, by saying: "I want to find the answer Z. If I knew Y I could find Z. If I knew X I could find Y . . . " and so forth until you get back to something you are given in the problem statement.
Some students find it useful to make a list of the information given and the goal to be uncovered (e.g. "given the constant acceleration, the initial velocity, and the time, find the displacement"). Others find it sufficient to write down only the goal (e.g. "to find: displacement").
Ineffective strategy. Do not page through your book looking for a magic formula that will give you the answer. Physics teachers do not assign problems in order to torture innocent young minds . . . they assign problems in order to force you into active, intimate involvement with the concepts and tools of physics. Rarely is such involvement provided by plugging numbers into a single equation, hence rarely will you be assigned a problem that yields to this attack. In those rare instances when you do face a problem that can be solved by plugging numbers into a formula, the most effective way to find that formula is by thinking about the physical principles involved, not by flipping through the pages in your book.
Make the problem more specific. You're asked to find the number of ways that M balls can be placed into N buckets. Suppose you can't even begin to map out a strategy. Then try the problem of 3 balls in 5 buckets. Solving the more specific problem will give you clues on how to solve the more general problem. And once you use those clues to solve the more general problem, you can check your solution by trying it out for the already-solved special case M=3 and N=5.
Large problems. At times you will be faced with big problems for which no method of solution is immediately apparent. In this case, break your problem into several smaller sub-problems, each of which is simple enough that you know how to solve it. At this strategy-design stage it is not important that you actually solve the sub-problems, but rather that you know you can solve them. You might begin by mapping out a strategy that leads nowhere, but then you haven't wasted time by implementing this strategy. Once you have mapped out a strategy that leads from the given information to the answer, you can then go back and execute the calculations. This strategy has been known from the time of the ancients under the name of "divide and conquer".
Execution (Tactics): Eventually, of course, you do have to roll up your sleeves and tinker with the problem. As you do so, keep your strategy in mind, and keep the following tips in mind as well:
Work with symbols. Depending on the problem statement, the final answer might be a formula or a number. In either case, however, it's usually easier to work the problem with symbols and plug in numbers, if requested, only at the very end. There are three reasons for this: First, it's easier to perform algebraic manipulations on a symbol like "m" than on a value like "2.59 kg". Second, it often happens that intermediate quantities cancel out in the final result. Most important, expressing the result as an equation enables you to examine and understand it (see the section on "Answer Checking") in a way that a number alone does not permit.
(Working with symbols instead of numbers can lead to confusion as to which symbols represent given information and which represent unknown desired answers. You can resolve this difficulty by remembering--as recommended above--to "keep the goal in sight".)
Define symbols with mnemonic names. If a problem involves a helium atom colliding with a gold atom, then define mh as the mass of the helium atom and mg as the mass of the gold atom. If you instead pick the symbols m1 and m2, you stand a good chance of mixing up the symbols and their meanings as you solve the problem. And if you don't define the symbols at all, but just begin throwing around m's and M's, you'll confuse both yourself and whoever is grading your answer.
Keep packets of related variables together. In acceleration problems, the quantity (1/2)at2 comes up over and over again. This collection of variables has a simple physical interpretation, transparent dimensions, and a convenient memorable form. In short, it is easy to work with as a packet. Take advantage of this ease. Don't artificially divide this packet into pieces, or write it in an unfamiliar form like t2a/2. Packets like this come up in all aspects of physics--some are even given names (e.g. "the Bohr radius" in atomic physics). Look for these packets, think about what they are telling you, and respect their integrity.
Image Credit: martinofranchi
