2015 AP Physics 1 free response 3b

The spring is now compressed twice as much to Δx = 2D. A student is asked to predict whether the final position of the block will be twice as far at x = 6D. The student reasons that since the spring will be compressed twice as much as before, the block will have more energy when it leaves the spring, so it will slide farther along the track before stopping at position x = 6D.

So let's just think about what the student is saying or what's being proposed here. In the first version, the first scenario, we compressed the spring by D, and then the spring accelerates the block. Then, right when we get back to x = 0, all of that potential energy has been turned into kinetic energy. The frictional force is acting against the motion of the block, so you can view it as it's providing negative work. The direction of the force is opposite to the change in x, and that negative work eventually causes the block to stop. All of that kinetic energy has now turned into heat, and so the block goes to 3D.

Now, in this new scenario, we could call that scenario two. We are going to compress the spring twice as far, so now we're going to compress the spring by 2D. This is x = -2D here. What's being said, or what's being proposed by the student, is all right, if we compress it twice as far, all of this potential energy is then going to be, we’re definitely going to have more potential energy here because it takes more work to compress the spring that far. All of that more potential energy is going to be converted to more kinetic energy once we get back to x = 0. So not only will it go further, but they're saying it'll go exactly twice as far. Instead of going to 3D, we're now going to go to 6D.

Let’s see what the questions are here. Part B1 says, which aspects of the student's reasoning, if any, are correct? Explain how you arrived at your answer. Then part two says, which aspects of the student's reasoning, if any, are incorrect? Explain how you arrived at your answer.

The student is correct that compressing the spring more will result in more energy when the block leaves the spring. This will result in the block going farther, which will result in a longer stopping distance.

Now, part two here: which aspects of the student's reasoning, if any, are incorrect? Let me see. Compressing twice as much does not result in exactly twice the stopping distance. We can explain this because we know that potential energy is equal to 1/2 times the spring constant times how much we compress squared. Energy grows with the square of compression. So, two times the compression will result in four times the energy. This will result in four times the stopping distance.

I think that goes a decent job of explaining where the student is correct and where their reasoning is incorrect.