2015 AP Physics 1 free response 3c

All right, now let's tackle part C. Use quantitative reasoning, including equations as needed, to develop an expression for the new final position of the block. Express your answer in terms of D.

All right, I'm going to set up a little table here for part C. Whoops, sorry about that; sometimes my pen is not functioning properly.

All right, so part C. Let me set up two scenarios.

So I have scenario one, scenario one, where we compress the spring. Our Delta X is equal to D. Then we have scenario two, scenario two, where we compress the spring by twice as much, is equal to 2D.

Let me set up my table now, so just like that and like this. And let's just think about a few things.

So the first thing I want to think about is the potential energy. So the potential energy when the spring is compressed, so potential potential energy when spring compressed—well, in this scenario, I'll call it potential energy for scenario 1. It's equal to 2 times the spring constant times how much we compress it squared.

Now, what about scenario two? This potential energy is going to be equal to 12 times the spring constant times how much we compress it—it's now twice as much squared.

Well, this is equal to 2 times the spring constant times 4D². I could put the four out front. This is equal to 4 * 12, 4 * 1 time our spring constant time D², which is equal to four times the potential energy when we just compressed the spring by D.

So, we already see a little bit of what we talked about in Part B. You compress your spring twice as much; you're going to have four times the potential energy because the potential energy doesn't grow proportionately with how much you compress it. It grows with the square of how much you compress it.

All right, now let's think about kinetic energy. Kinetic kinetic energy when X is equal to zero—so right when the spring, when we lose contact with the spring, the spring is no longer pushing on the block. Well, our kinetic energy is going to be equal to what our potential energy was when the spring was actually compressed.

Or another way of thinking about it; all of that potential energy has now been turned into kinetic energy. Now, what about over here? Well, the kinetic energy in this scenario, like we just saw before, that's going to be equal to the potential energy when the spring was compressed, or all of that potential energy gets turned into kinetic energy.

And this is equal to four * U1, four times the potential energy in scenario one, which is the same thing as four times the kinetic energy in scenario one. So we have four times the kinetic energy, four times kinetic energy, kinetic kinetic energy.

So then we have stopping distance. Stopping stopping distance. We know here this is 3D.

And then we know—then we can say, well, what is this question mark? Well, let's just think a little bit about this. We know that if we have that kinetic energy at x equals 0, so we know that K1 plus the work done by friction—so let me make it clear, this right over here, that is work done by friction, work done by friction—it's going to be negative work because the force of friction is acting in the direction opposite of the change in X.

So the kinetic energy plus the work done by friction is going to be equal to zero. This work cancels out; all of this energy, it's one way to think about it, is turning it all into heat.

And so let's think about what the work done by friction is equal to. Well, the work done by friction is equal to the coefficient of friction times the mass of the block times the gravitational field times how far that, how over what distance that force—this right over here is a force of friction—times over what distance that force was applied, so times 3D.

And to be clear, this force is going in the opposite direction of our change in X. So because of that, this will be negative.

And so we can say, we can say that the kinetic energy at x equals 0—and now I can just write it as minus μ, the coefficient of friction, times mass, times a gravitational field, times 3D—is equal to zero.

We can add this to both sides, and we get K1 is equal to μ * m * g * 3D.

And if you wanted to solve for distance here, you can divide both sides by the force of friction—so divide both sides by μ * m * g—and you get 3D.

And I'm just swapping the sides here. It's going to be equal to the amount of kinetic energy we have right at x equals 0 divided by μ * m * g.

And you could just view this as the force of friction—the force, I'll just call it the force of friction right over there.

So if you want to figure out your stopping distance, you just figure out your kinetic energy right when you start entering into the friction part of your platform, and then you divide that by the force of friction, and that will give you your distance traveled.

So the distance here, distance—so I can just put some arrows right over here—our distance is going to be equal to K2 divided by the force of friction.

Well, K2 is equal to four * K1, is equal to four times K1, and our force of friction is going to be the same. We have the same coefficient of friction, we have the same mass, we have the same gravitational field.

So divided by the force of friction—and this we already know. K1 divided by the force of friction is equal to 3D.

So this is all going to be equal to four. This is going to be four times 3D, which is equal to 12D.

So this is all a mathematical way of saying you compress it twice as much, you're going to have four times the potential energy when your spring is compressed, which means you're going to have four times the kinetic energy at x equals 0, which means it's going to take, you're going to have four times the stopping distance.

So instead of stopping at 3D, or in 6D is what the student proposed, you are now stopping at 12D. So that is our stopping distance.

Did we answer all of—yeah, we answered all of part C. Use quantitative reasoning, including equations as needed, to develop an expression for the new final position of the block. Express your answer in terms of D. Yep, feel good about that.