Question 3a: 2015 AP Physics 1 free response

- [Voiceover] A block is initially at position x = zero, and in contact with an uncompressed spring of negligible mass. The block is pushed back along a frictionless surface from position x = zero to x = -D, as shown above, compressing the spring by an amount delta x = D. So, the block starts here, and it's just in contact with the spring, so it's initially, the spring is uncompressed. And it's just touching the block. And 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 = to D, so we compress, we move this block back over to the left by D, that compresses the spring by D. The block is then released at x = zero, the block enters a rough part of the track and eventually comes to rest at position x = 3D. So when we compress the spring, we're actually doing to work to compress the spring, so that work, that energy from the, or the work we're doing, gets stored as potential energy in the spring-block system. And 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 = zero, then the spring is back to uncompressed, so it's not gonna keep pushing on the block after that point. And then the block's going to have this kinetic energy and if there was no friction in this gray part here, it would just keep on going forever. And if there's no air resistance, and we're assuming no air resistance for this, for this problem, but since there is friction, it's just going to decelerate it 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 = 3D, the coefficient of kinetic friction between the block and the rough track is mu. Alright, on the axes below, sketch and label graphs of the following two quantities as a function of the position of the block between x = negative D and x = 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. So 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. So you have to think about what is the potential energy of a compressed spring? Well, the potential energy the potential energy is equal to one-half times the spring constant times how much you compress the spring squared. So if we wanna 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, the potential energy of a compressed spring or the work necessary to compress a spring, cause the work necessary to compress the spring that's going to be the potential energy that you're essentially putting into that system. And so, for this, as we compress the spring to D, you are, you're going to end up with a potential energy of one-half times the spring constant x our change in x is D, our change in x is D. x D, x D squared. So let's plot that on this right over here. So right, whoops, right when we are at x = zero there's no potential energy in our system, but then we start to compress it, and when we get to x = 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, let's 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's going to be like once we've compressed the spring by D. And it's not going to be a linear relationship, remember the potential energy potential energy is equal to one-half times the spring constant, times the spring constant, times how much you've compressed the spring squared. So, the potential energy increases as a sqaure of how much we've compressed the spring. So when we've compressed the spring half as much, we're going to have one-fourth of the potential energy. So it's going to look like this, it's gonna be you can view it as the left side of a parabola. So it's going to, going to look something, something like this. So that's the potential energy. Now, when you're in 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, as the spring accelerates the block, you're gonna go down this potential energy curve, as you go to the right, but then, it gets converted to kinetic energy. So the potential energy plus the kinetic energy needs to be constant, at least over this period from x = negative D to x = zero. So the kinetic energy starts off at zero, it's stationary, but then, it starts, the block starts getting accelerated. It starts getting accelerated. And the sum, the sum of these two things needs to be equal to one-half times our spring constant times D squared. And so you can see if you, if you were to add these two curves at any position, you are going, their sum is going to sum up to this value. And so right when you get back to x = zero, all of that potential energy has been converted into kinetic energy. And then that kinetic energy, we would stay at that high kinetic energy if there was no friction or no air resistance. But we know that the block comes to a rest at x is = to 3D. So all the kinetic energy is gone at that point, and you might say well, what's that getting converted into? Well, it's gonna get converted into, into heat, due to the friction. So that's where, ya know, energy cannot be cannot be created out of thin air or lost into thin air, it's converted from one form to another. And 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? And the key realization is: 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, and the mass of the block isn't changing, so the force of friction's going to be the same. And it's acting against the motion of the block. So you can, 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. And so, this is going to decrease at a linear rate. So, let me draw that. So it's gonna be a linear decrease, just like that. And the key thing to remind yourself is: is 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, and 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, kinetic energy, and in blue, just to make sure I label it right, this is the potential energy, potential, potential energy.