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- [Voiceover] Alright,
now let's tackle part c.

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Use quantitative reasoning,
including equations as needed,

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to develop an expression

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for the new final position of the block.

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Express your answer in terms of D.

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Alright, I'm gonna set up a
little table here for part c.

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Oops, sorry about that.

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Sometimes my pen is not
functioning properly.

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Alright so part c.

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Let me set up two scenarios.

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So I have scenario one.

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Scenario one, where we
compress the spring,

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our delta x is equal to d,

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and then we have scenario two,

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scenario two, where we
compress the spring by

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twice as much, is equal to two d,

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and let me set up my table now,

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so, just like that and like this,

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and let's just think about a few things.

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So the first thing I wanna think about is

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the potential energy,
so the potential energy

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when the spring is compressed.

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So potential energy
when spring compressed.

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When, I'll just write
compressed, when compressed.

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Well in this scenario I'll
call it the potential energy

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for scenario one, it's
equal to one half times

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the spring constant, times how
much we compress it squared.

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Now what about scenario two?

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This potential energy
is going to be equal to

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one half times the spring
constant, times how much

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we compress it, it's now
twice as much, squared,

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well this is equal to one half
times the spring constant,

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times four d squared,

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and I can out the four up front,

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this is equal to four times
one half, four times one half,

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times our spring
constant, times d squared,

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which is equal to four
times the potential energy

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when we just compress the spring by d.

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So we already see a little
bit of what we talked about

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in part b, you compress
your spring twice as much,

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you're going to have four
times the potential energy,

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because the potential energy
doesn't grow proportionately

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with how much you compress
it, it grows with the square

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of how much you compress it.

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Alright, now let's think
about kinetic energy

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Kinetic energy when x is equal to zero,

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so right when the spring,
when we lose contact

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with the spring, the
spring is no longer pushing

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on the block, well our
kinetic energy is going to be

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equal to what our potential
energy was when the spring

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was actually compressed.

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Another way of thinking about
it, all that potential energy

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has now been turned into kinetic energy.

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Now what about over here?

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Well, the kinetic energy in this scenario,

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like we just saw before,
that's gonna be equal to the

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potential energy when the
spring was compressed.

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All of that potential energy
gets turned into kinetic energy

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and this is equal to four
times u one, four times

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the potential energy in
scenario one, which is the same

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thing as four times, which
is equal to four times

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the kinetic energy in scenario one.

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So we have four times the kinetic energy.

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Four times kinetic energy,
kinetic, kinetic energy.

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So then we have stopping
distance, stopping,

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stopping distance.

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We know here this is three d and we know,

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and then we can say "What's
this?", this question mark.

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Well let's just think a
little bit about this.

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We know that if we have
that kinetic energy

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at x equals zero,

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so we know that k one plus
the work done by friction,

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so let me make it clear,
this right over here,

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that is work done by friction,
work done by friction,

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and it's gonna be negative work
cause the force of friction

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is acting in the direction
opposite of the change in x,

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so the kinetic energy plus
the work done by friction

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is going to be equal to zero.

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This work cancels out all of this energy.

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When we think about it, it's
turning it all into heat.

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And so let's think about what
the work done by friction

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is equal to.

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Well the work done by friction
is equal to, is equal to,

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the coefficient of friction
times the mass of the block,

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times the gravitational
field, times how far,

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over what distance that
force, this right over here

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is the force of friction,
times over what distance

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that force was applied, so times three d.

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And to be clear, this force is going in

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the opposite direction of our change in x,

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so because of that this
will be a negative,

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and so we can say, we can
say that the kinetic energy

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at x equals zero, and now I
can just write it as minus

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mu, the coefficient of
friction, times mass, times

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the gravitational field, times
three d is equal to zero.

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We can add this to both
sides and we can get

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k one is equal to mu times
m times g times three d,

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and if you wanted to
solve for distance here,

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you can divide both sides
by the force of friction.

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So divide both sides by mu
times m times g, and you get

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three d, and I'm just
swapping the sides here,

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is going to be equal to the
amount of kinetic energy we have

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right as x equals zero,
divided by mu times m times g

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and you can just view this
as the force of friction.

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The force, I'll just call it

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the force of friction right over there.

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So if you wanna figure
out your stopping distance

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you just figure out your kinetic energy,

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right when you, right at x equals zero,

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right when you start entering
into the frictiony part of

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your platform and then you divide that by

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the force of friction,
and that will give you

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your distance traveled.

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So the distance here, distance,
so I can just put some

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arrows right over here,
our distance is going to be

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equal to k two divided by the force of,

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the force of friction.

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Well k two is equal to four times k one,

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is equal to four times k one.

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and our force of friction
is going to be the same

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We have the same coefficient of friction,

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we have the same mass, we have the same

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gravitational field, so
divided by force of friction,

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and this we already know, k one
divided by force of friction

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is equal to three d, so this
is all going to be equal to

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four, this is going to
be four times three d,

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three d, which is equal to 12 d, 12 d.

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So this is all a
mathematical way of saying

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you compress it twice as
much, you're going to have

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four times the potential energy when your

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spring is compressed, which
means you're going to have

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four times the kinetic
energy at x equals zero,

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which means it is going to
take, you're going to have

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four times the stopping
distance, so instead of stopping

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at three d or in six d as
what the student proposed,

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you are now stopping at 12 d, so that is

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our stopping distance.

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Did we answer all of, yeah
we answered all of part c.

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Use quantitative reasoning,
including equations as needed,

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to develop an expression
for the new final position

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of the block.

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Express your answer in terms of d.

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Yep, feel good about that.