General guide to answering AP physics free response questions

1. Read the question carefully.

This should go without saying, but I have met a lot of rushed (or lazy) students who just won't read the entire question.

If you have trouble keeping track of the information, underline the important pieces of information and circle what you're supposed to find. This should help if you get confused about what you are supposed to be doing.

2. Draw a picture and FBD.

Drawing a picture should be automatic for any free response question. Why? A picture does several things:

(a) It helps you understand exactly what is happening.

Can you transform a text description into a clear mental image of what is happening in the system? Are you forgetting some aspect of complexity? Are you confident you will remember whether something is frictionless or not once you start working?

If you can answer all of the above "yes", then congratulations. Your memory and reading comprehension are spectacular. You might not need to draw a picture. But for the rest of us (not them, US), a picture can help us really understand what's happening, or, perhaps as importantly, that we don't quite understand it and need to read the statement again more carefully.

(b) It helps trigger memory for similar problems.

The brain stores images as well as words. Your brain might not process a paragraph of text as similar to other bits of text floating in your head. But once you draw a picture, you might better form the connection. A picture of an object/skiier/car going down a hill might help you remember that you have solved these problems before, and possibly even help you recall what methods you should use.

(c)  It provides a way of storing some important information.

Even if you underline important pieces of information in the problem statement, you might find it easier to consoldiate all the really important bits in a picture. Instead of wondering, "what angle is that?", you can indicate it directly on the picture. It's a good way of keeping track of forces and their directions. In fact, I'd encourage a division of labor -- use the diagram to keep track of directions and coordinates, and a short list (see #3 below) to keep track of numerical quantities.

FBD: In addition to a picture of the system, construct a free body diagram for each object of interest. In other words, draw a free body diagram for every object that has motion you are asked to describe. For problems involving stacked blocks, you might need to construct a FBD for each block.

A FBD isn't needed for every problem, but it's a good idea for many of them, even problems that don't directly ask for it. It's basically mandatory for all problems using Newton's Second Law.

Note: Include a coordinate system/sign convention on both your system picture and the FBD.
3. List important quantities and the variables you use to represent them.

This is important for a couple reasons.

(a) Not all quantities have a universally accepted symbol. You don't have to define "g". But what about drag force? Do you use "$f$", or "$f_{drag}$"?

(b) It helps to organize information. As stated above, you can put some information on a diagram. But it will get very cluttered if you put every pertinent number on that diagram. I recommend keeping the directional information on the diagram, and the numerical information/defintion of variables in a short list below the picture.

(c) This helps you keep track of pieces of information that aren't explicitly stated. Statements like "starting from rest" indicate that $v_0 = 0$. Otherwise, it might be possible to overlook the quantitative information in what appears to be a piece of the problem statement text devoid of numbers.

(d) For some problems, it can help you narrow down what type of problem it is. Does it have a radius and an angular velocity? Then it probably involves rotational mechanics. For simple kinematics problems, it can help you pick which of the four equations to use. If your problem gives an initial and final velocity, an acceleration, and wants you to find a displacement, then you probably are best served using $v^2 = v_0^2 + 2a\Delta x$.

4. Diagnose the type of problem.

Yes, you do all the prep work of #1-3 first. That's because you do not want to jump in and start writing a bunch of random equations. Unless you know precisely what you're doing, you will confuse yourself.

Now, how do you diagnose which problem? If you've read the problem carefully, drawn a picture, and listed the pertinent information, then you probably have enough to tell the difference between a kinematics problem, a spring problem, a rotational mechanics problem, a drag force problem, a gravitation problem, etc.

Now the FRQs are multi-step, and integrate many different concepts into each problem. But a given step (and especially the first step) might use only one or two concepts.

This is the step where things can go wrong. But it won't if you have enough practice, training and experience. There really is no substitute for a solid conceptual background and lots of practice solving problems.

If it's a problem dealing with a spinning rod, and you don't know anything about rotational mechanics, then you won't be able to solve this problem. If you know a little about rotational mechanics, but don't know anything about a section critical for this problem (parallel axis theorem, or rotational kinematic equations, or torsional coefficients), then you might get a little farther, but not far enough. If you know conceptually all the topics, but don't have a lot of experience actually doing the free response questions, you might not get full credit if you derivations are a huge mess and you don't know the difference between "Derive, Explain, Find, Solve, and Justify" when used in one of these questions. (These keywords in the problem statement give you a sense of what kind of work you need to show in order to get full credit.)

5. Solve the problem.

If #1-4 were done properly, then this will actually be pretty easy.  A student with a solid background and plenty of practice can do steps #1-4 in about 2 minutes. This means that they have 12 minutes to solve the problem. That should be more than enough time.

More specifically, this means different things for different problems.

For graph interpretations, it might mean determining the functional form of the graph (sine, cosine, -sine, -cosine).

For many problems, it will start by choosing an appropriate equation. Once you've diagnosed the problem, and have enumerated the important quantities, then it should be a matter of choosing between a handful of equations, some of which are given to you on the formula sheet!

For some problems, it asks for a qualitative judgment. Here, again, pictures help, as well as some rough calculations.

Some problems may ask for an answer based on a limiting condition. For example, at terminal velocity, the object's acceleration is zero, which means, by Newton's Second Law, that the sum of the forces on the object is zero.

General guidelines for solving the problem:

(a) Sketch a chain of steps you need to take in order to get the answer. For example, let's say you have a roller coaster problem involving a loop track, asking you to find the normal force at the top of the loop. Your chain might look like this: solve for total energy of cart before loop $\rightarrow$ calculate  change in potential energy of cart at top of loop $\rightarrow$ use conservation of energy to find velocity at top of track $\rightarrow$ find centripedal force$\rightarrow$ subtract gravity from centripedal force to get normal force.

Some of you might want to skip this, if you are confident you can "see" the chain of steps without writing it down in words. Others might prefer writing it, in at least abbreviated form. I'll leave writing this as optional, but thinking about this before you start working is absolutely mandatory.

(b) When writing, be neat and tidy. For those of us with bad handwriting, it stinks to confuse a 7 for a 1 (or even a 5 for a f, 2, or 8 -- it happens). Take extra space if you need to -- you can always state clearly where your work is continued.

(c) Take the time to show the key steps in your derivation. Skip steps at your own risk. You don't need to explicitly show that you divided by a variable on both sides. You shouldn't show every step of algebra that you were required to show in a 7th grade class. That said, if you skip too many operations between two lines of work, you increase the risk of making an error.

(d) If you get lost and confused, stop. Go back to (a) and think about whether you are doing the right step, or even one that's necessary given the problem. Think it through conceptually. This will save you time and stress, and probably give you more points. (It's easy to get lost, then stuck, if you start manipulating equations without stopping to think about the process.

(e) Box your final answer and, if necessary, provide directions that indicate the flow of your work. This helps the grader find it. This is really important if your workspace is a bit chaotic, with bits floating in the margins and wrapping around. Although it's HIGHLY recommended your work be organized into at most two columns, left to right, arrows and directions are acceptable if absolutely necessary.

(f) Use conversion factors to convert between units. Although it's unlikely you'll need them (generally, there is SI unit consistency within the exam), use conversion factors to make absolute certain you are properly converting between units.

6. Check your answer.

There are quite a few checks you can make. Not all will apply in a given problem, but at least one should be valid.

(a) Does it have the right units?

If the units are wrong, you might have dropped them. Alternatively, maybe you forgot to multiply or divide by a critical factor. Units aren't just a way of bookeeping -- an error in units can help you trace precisely where you made a mistake. If I should end up with $m/s$ and end up with $m^2/s^2$, then I probably forgot to take a square root. If I should have $kg\ m/s$ and instead have $m/s$, I might've forgotten to multiply by mass in a momentum calculation.

(b) Does it have the right sign? 

The wrong sign could indicate a serious conceptual error. Maybe you didn't set up your coordinate system/sign convention properly. Or maybe you subtracted when you were supposed to add. Another common mistake is to not remember that the solution of $x^2 = k$ is $x = \pm \sqrt{k}$. You may actually need to choose either the positive or the negative root in a given situation -- and which one is correct will be set by the constraints of the problem.

For example, in a kinematics problem solved by using $x = x_0 + v_0 t +\frac{1/2}at^2$, the solution for $t$ will usually involve an expression that includes $\pm\sqrt{\rm something}$. In some cases, choosing the negative root leads to a negative time. If the constraints of the problem indicate that motion is considered for $t\geq 0$, then that means the solution with the $-\sqrt{\rm something}$ should be discarded.

(c) Does the magnitude make sense?

Do you get something falling with air resistance that has acceleration with magnitude $> g$? Then you probably have the wrong sign/direction on the drag force. Do you get a mass for a planet roughly equal to the mass of a person? Then you probably made a multiplicative error/forgot to square something.

(d) If you take a limiting case, do you get an answer that makes sense?

For example, if the tension in a pulley system goes to zero, does the acceleration of a hanging block go to $g$? (In other words, if you cut the cord, does the solution reflect that the block will experience freefall?) Does your complicated solution for the velocity of an object subject to air resistance reduce to the terminal velocity if $t \rightarrow \infty$?

*****

In the next post, I actually show you how to apply this to a problem. A good framework isn't worth anything if you don't see it applied and understand how it works.