2015 AP Physics 1 free response 3a

A block is initially at position x equals zero and in contact with an uncompressed spring of negligible mass. The block is pushed back along a frictionless surface from position x equals zero to x equals negative d, as shown above, compressing the spring by an amount delta x equals d. So, the block starts here and it's just in contact with the spring. Initially, the spring is uncompressed and it's just touching the block. Then we start to compress the spring by pushing the block to the left, and we compress it by an amount d. They tell us that right there, delta x is equal to d.

So, we compress, we move this block back over to the left by d, and that compresses the spring by d. The block is then released at x equals zero. The block enters a rough part of the track and eventually comes to rest at position x equals 3d. So, when we compress the spring, we're actually doing work to compress the spring, and so that work, that energy from that, or the work we're doing, gets stored as potential energy in the spring-block system. Then, when we let go, that potential energy is going to be converted to kinetic energy, and that block is going to be accelerated all the way until we get back to x equals zero.

Then the spring is back to uncompressed, so it's not going to keep pushing on the block after that point. The block is going to have this kinetic energy, and if there was no friction in this gray part here, it would just keep on going forever. If there's no air resistance—and we're assuming no air resistance for this problem—since there is friction, it's just going to decelerate at a constant rate. You're going to have a constant force of friction being applied to this block.

So, let's see, they say they tell us that it's going to come to rest at x equals 3d. The coefficient of kinetic friction between the block and the rough track is mu. Alright, on the axis below, sketch and label graphs of the following two quantities as a function of the position of the block between x equals negative d and x equals 3d. You do not need to calculate values for the vertical axis, but the same vertical scale should be used for both quantities.

So, they have the kinetic energy of the block and the potential energy of the block-spring system. Let's first focus on the potential energy u because when we start the first part of this, when we're compressing the spring, that's when we're starting to put potential energy into this spring-block system.

You have to think about what the potential energy of a compressed spring is. Well, the potential energy is equal to one-half times the spring constant times how much you compress the spring squared. If we want to say delta x is how much you compress the spring, that squared. Now, if what I just wrote is completely unfamiliar to you, I encourage you to watch the videos on Khan Academy about the potential energy of a compressed spring or the work necessary to compress a spring, because the work necessary to compress the spring is going to be the potential energy that you're essentially putting into that system.

So, for this, as we compress the spring to d, you're going to end up with a potential energy of one-half times the spring constant times our change in x, which is d, times d, times d, the squared. So, let's plot that over here. Right—whoops! Right when we are at x equals zero, there's no potential energy in our system, but then we start to compress it, and when we get to x equals d, we're going to have a potential energy of one-half times the spring constant times d squared.

So, let's just say this right over here. Let's say that over there—actually, let me do a—see that one is actually—I’ll do it over here so it’ll be useful for me later on. So, let's say that this right over here is one-half times our spring constant times d squared.

So, this is what our potential energy is going to be like once we've compressed the spring by d. It's not going to be a linear relationship. Remember, the potential energy is equal to one-half times the spring constant times the spring constant times how much you've compressed the spring squared. The potential energy increases as a square of how much we compress the spring. So when we've compressed the spring half as much, you're going to have one-fourth the potential energy. So, it's going to look like this. You could view it as the left side of a parabola.

So, that's the potential energy. Now, when you're at this point, when the thing is fully compressed and then you let go, what happens? Well, that potential energy is turned into kinetic energy. So, as the spring accelerates the block, you're going to go down this potential energy curve as you go to the right, but then it gets converted to kinetic energy.

Thus, the potential energy plus the kinetic energy needs to be constant, at least over this period from x equals negative d to x equals zero. The kinetic energy starts off at zero; it's stationary, but then the block starts getting accelerated. It starts getting accelerated, and the sum of these two things needs to equal one-half times our spring constant times d squared.

You could see if you were to add these two curves at any position, their sum is going to sum up to this value. Right when you get back to x equals zero, all of that potential energy has been converted into kinetic energy. Then that kinetic energy would stay at that high kinetic energy if there was no friction or air resistance, but we know that the block comes to rest at x equals 3d.

So, all the kinetic energy is gone at that point. You might say, "Well, what's that getting converted into?" Well, it's getting converted into heat due to the friction. So, that's where energy cannot be created out of thin air or lost into thin air; it's converted from one form to another.

So, the question is, what type of a curve is this? Do we just connect these with a line, or is it some type of a curve? The key realization is that you have a constant force of friction the entire time that the block is being slowed down. The coefficient of friction doesn't change, so the force of friction—or the mass of the block—isn't changing.

The force of friction is going to be the same, and it's acting against the motion of the block. You can view the friction as essentially doing this negative work, and so it's sapping the energy away. If you think about it relative to distance in a given amount of distance, it's sapping away the same amount of energy; it's doing that same amount of negative work.

This is going to decrease at a linear rate. So, let me draw that. It’s going to be a linear decrease, just like that. The key thing to remind yourself is that this is a plot of energy versus position, not velocity versus position or velocity versus time or energy versus time—this is energy versus position. That’s what gives us this linear relationship right over here.

So, we have the kinetic energy (K) of the block—that's what I did in magenta—so this is the kinetic energy. Kinetic energy. And in blue, just to make sure I label it right, this is the potential energy. Potential energy.