Tackle physics problems like a gymnast: spot your answer then stick the landing!

Welcome back!  Let me start this post with a story and a few questions.  I was working through mechanics problems with a student I was tutoring in physics.  I sketched out the next problem, turned to him, and asked, “Alright, now what does the answer look like?”  We had not even begun to discuss the problem, and the question took him somewhat by surprise.

Why did I ask what the answer would be, and why did my question surprise him?  Most importantly, how can my question improve the way you approach physics problems?

When you approach a problem, think about the answer you expect to find.

In my tutoring, I find that students often skip this crucial step.  They begin to work the problem, to lay out their mathematical tools and begin chipping away at the stone block of the question, before they have formulated a clear idea of the goal they are trying to achieve.  If you take the time to think carefully about the problem before you have written your first equal sign, you can often form a pretty good guess for what your answer should be.  This guess can help guide your problem-solving strategy and can function as a consistency check when your answer emerges from the math.

Think of each question as an experiment: you are presented with a question and it is your job as a scientist to find the answer.  A crucial part of every experiment is forming the initial hypothesis.  We rarely embark upon an experimental project without having a clear goal in mind from the outset.  If we carefully confirm our hypothesis, that’s great.  If we disprove our hypothesis, that can be even more useful.  We double-check our work, consider how our initial thinking might have been flawed, and deepen our understanding.

It can be easy to form a rough guess!

We need not put much thought into formulating our guess; the effort will come when we work through the math.  In order to see how we might intuit the answer with a careful appraisal of the problem, let’s consider the question I posed to my student.

A glass of mass m is sitting on a table.  The coefficient of static friction between the glass and the tablecloth is μ.  How quickly must you pull out the tablecloth so that the glass does not move?

Let’s brainstorm some of the questions that might go through your mind and the answers that might help you form your guess:

• What are they looking for?  What kind of physical quantity is it?  “How quickly”…that’s velocity or acceleration.  The tablecloth is starting from rest and then you yank it out.  They must be looking for acceleration.
• Am I looking for one exact answer, or for an upper or lower limit?  In other words, what kind of equality sign should my final answer have?  Let’s think about what happens if you pull the tablecloth out infinitely quickly.  The glass will not move.  If you pull very slowly, on the other hand, the glass will move.  That says we’re looking for a lower limit for the acceleration – the glass will move if you pull any more slowly than this critical acceleration and the condition won’t be satisfied, but the glass will not move if you pull any more quickly than this critical acceleration and the condition will be satisfied.  I want an equation that looks like “a > …” or “a ≥ …”   I don’t know which one it will be – will the glass move if you pull precisely at the critical acceleration? – but we need not worry about that detail when forming our guess.
• What physical quantities am I given as inputs?  The problem gives me m and μ.  Gravity is holding the glass to the tablecloth and is a physical constant, so I also implicitly have g to work with.
• How do I use these inputs to find an acceleration for my answer?  The only acceleration I’m given is g.  Just from considering our units, the answer must be something like “a > … g.”
• How about the other inputs?  Does the mass matter?  There are no units of mass in the answer.  Either the mass of the glass enters as part of a ratio so that its units cancel out – for example, when you use conservation of momentum to calculate the velocities of objects after a collision, the answer will often take the form 1 = m1/m2 v2 or something similar – or the mass doesn’t appear in the answer at all.
• Is there any other relevant mass in the problem I could use to cancel out the units of mNo, there doesn’t seem to be any other mass.  The mass of the tablecloth doesn’t matter because we’re looking for the acceleration with which you pull it, not the force.  It seems like the mass of the glass does not appear in the answer.
• Really, the mass doesn’t matter at all?  Does that make sense?  Well, where might mass appear?  It will help determine how much force is needed to accelerate the glass through F=ma, and it will also appear in the force of friction.  It’s plausible that these two appearances will cancel out.
• How about μ?  Will that factor into the answer?  Units don’t help us here because μ does not have units, so let’s use our physical intuition.  Would it matter if the tablecloth were stickier?  Would it matter if you used a sticky rubberized outdoor tablecloth or the kind of smooth cotton tablecloth you see in a fine restaurant?  It seems like it would be harder to prevent the glass from moving on the stickier tablecloth, so μ should appear in the answer.
• How should μ appear?  Units don’t tell us if the answer should look like g/μ or μg.  If we increase μ (make the tablecloth sticker), we’ll need a higher acceleration.  The answer should look like μg.

There we have our guess!  Our answer should look like “a > μg.”  As it turns out, that is the answer exactly!

I know that you probably could have done the math in less time than it took you to read my imagined musings on the problem, but faster appraisals come with practice.  Once you start asking these questions for every problem, the answers will come faster and more easily.  Eventually, you’ll be able to form a decently educated guess in a few seconds.  Your equation solving skills will not speed up so dramatically!

A guess is not an answer!  You still need to solve the problem.

You must be careful not to confuse your guess with a rigorous answer to the problem!  We used very simple arguments to predict roughly what the answer should look like.  We happened to be lucky and land on the exact answer in the previous example, but we could just as easily have missed a factor of 2 or some similar numeric factor.  You still need to solve the problem carefully using the methods you’ve learned in class, but now you have “a > μg” tucked away in the back of your mind to inspire your problem-solving strategy and to confirm that the answer you eventually come to is reasonable.

One important exception to this rule occurs when you are taking the physics SAT or physics GRE.  When you begin a problem, run through the logical process we sketched above and come up with your guess.  If your guess enables you to eliminate all but one of the answers with a few seconds’ worth of effort, that’s a huge win!  Even if you can only eliminate two or three of the answers, that’s still a very effective use of your time.

Beware of surprises in the math

Relying too heavily on this guessing strategy can be dangerous.  Again, it’s a tool to help you solve the problem quickly and with confidence; it’s not a solution in itself.

To demonstrate a potential hazard of the guessing strategy, let’s consider the next problem I posed to my student:

A rollercoaster car of mass m travels down a frictionless ramp and around a loop of radius R.  From what height h above the bottom of the loop must the car start in order not to fall off the track at the top of the loop?

Let’s run through an abbreviated version of the guessing process.  The only length in the problem is the radius R, so the answer must be something like “h = … R.”  The mass m and g both accelerate the car down the ramp and try to pull the car off the track at the top of the loop, so it’s plausible that they drop out of the answer.  There’s no friction, so we only care about the height of the ramp, not how far along the ramp the car travels.  Our guess is “h = R.

As it turns out, though, the exact answer is “h = 5/2 R.”  There’s no obvious way to guess the numeric factor of 5/2, so we need to solve the problem carefully in order to find the correct answer.  Still, though, the fact that the answer we found is close in form to the answer we expected to find boosts our confidence in our answer.

Guess early, guess often!

You should use this guessing strategy whenever you approach a new problem.  The more guesses you formulate and check, the stronger your physical intuition will become and the quicker and more accurate your guesses will be.  Try it out on your next homework, or if you're still struggling, consider getting in touch with a physics tutor!

Tags: physics