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# Question 3a: 2015 AP Physics 1 free response

## Video transcript

- [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.