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Welcome back.

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I'll now do another conservation
of energy

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problem, and this time I'll
add another twist. So far,

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everything we've been doing,
energy was conserved by the

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law of conservation.

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But that's because all of the
forces that were acting in

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these systems were conservative
forces.

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And now I'll introduce you to
a problem that has a little

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bit of friction, and we'll see
that some of that energy gets

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lost to friction.

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And we can think about
it a little bit.

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Well where does that
energy go?

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And I'm getting this problem
from the University of

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Oregon's zebu.uoregon.edu.

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And they seem to have some nice
physics problems, so I'll

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use theirs.

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And I just want to make
sure they get credit.

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So let's see.

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They say a 90 kilogram
bike and rider.

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So the bike and rider combined
are 90 kilograms. So let's

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just say the mass is 90
kilograms. Start at rest from

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the top of a 500 meter
long hill.

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OK, so I think they mean
that the hill is

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something like this.

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So if this is the hill, that
the hypotenuse here is 500

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hundred meters long.

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So the length of that,
this is 500 meters.

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A 500 meter long hill with
a 5 degree incline.

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So this is 5 degrees.

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And we can kind of just view
it like a wedge, like we've

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done in other problems.
There you go.

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That's pretty straight.

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OK.

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Assuming an average friction
force of 60 newtons.

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OK, so they're not telling us
the coefficient of friction

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and then we have to figure
out the normal

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force and all of that.

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They're just telling us, what
is the drag of friction?

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Or how much is actually friction
acting against this

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rider's motion?

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We could think a little bit
about where that friction is

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coming from.

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So the force of friction is
equal to 60 newtons And of

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course, this is going to be
going against his motion or

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her motion.

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And the question asks us, find
the speed of the biker at the

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bottom of the hill.

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So the biker starts up
here, stationary.

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That's the biker.

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My very artful rendition
of the biker.

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And we need to figure out the
velocity at the bottom.

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This to some degree is a
potential energy problem.

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It's definitely a conservation
of mechanical energy problem.

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So let's figure out what the
energy of the system is when

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the rider starts off.

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So the rider starts off at
the top of this hill.

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So definitely some
potential energy.

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And is stationary, so there's
no kinetic energy.

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So all of the energy is
potential, and what is the

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potential energy?

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Well potential energy is equal
to mass times the acceleration

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of gravity times
height, right?

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Well that's equal to, if the
mass is 90, the acceleration

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of gravity is 9.8 meters
per second squared.

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And then what's the height?

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Well here we're going to have
to break out a little

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trigonometry.

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We need to figure out this side
of this triangle, if you

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consider this whole
thing a triangle.

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Let's see.

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We want to figure out
the opposite.

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We know the hypotenuse and
we know this angle here.

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So the sine of this angle is
equal to opposite over

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hypotenuse.

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So, SOH.

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Sine is opposite over
hypotenuse.

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So we know that the height--
so let me do a little work

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here-- we know that sine of
5 degrees is equal to

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the height over 500.

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Or that the height is equal
to 500 sine of 5 degrees.

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And I calculated the sine of
5 degrees ahead of time.

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Let me make sure I
still have it.

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That's cause I didn't have my
calculator with me today.

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But you could do this
on your own.

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So this is equal to 500,
and the sine of

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5 degrees is 0.087.

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So when you multiply these
out, what do I get?

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I'm using the calculator
on Google actually.

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500 times sine.

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You get 43.6.

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So this is equal to 43.6.

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So the height of the hill
is 43.6 meters.

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So going back to the potential
energy, we have the mass times

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the acceleration of gravity
times the height.

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Times 43.6.

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And this is equal to, and then
I can use just my regular

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calculator since I don't
have to figure out

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trig functions anymore.

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So 90-- so you can see the whole
thing-- times 9.8 times

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43.6 is equal to, let's
see, roughly 38,455.

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So this is equal to 38,455
joules or newton meters.

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And that's a lot of
potential energy.

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So what happens?

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At the bottom of the hill--
sorry, I have to readjust my

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chair-- at the bottom of the
hill, all of this gets

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converted to, or maybe I should

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pose that as a question.

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Does all of it get converted
to kinetic energy?

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Almost. We have a force
of friction here.

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And friction, you can kind of
view friction as something

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that eats up mechanical
energy.

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These are also called
nonconservative forces because

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when you have these forces
at play, all of the

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force is not conserved.

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So a way to think about it is,
is that the energy, let's just

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call it total energy.

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So let's say total energy
initial, well let me just

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write initial energy is equal
to the energy wasted in

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friction-- I should have written
just letters-- from

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friction plus final energy.

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So we know what the initial
energy is in this system.

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That's the potential energy
of this bicyclist and this

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roughly 38 and 1/2 kilojoules
or 38,500 joules, roughly.

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And now let's figure out the
energy wasted from friction,

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and the energy wasted from
friction is the negative work

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that friction does.

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And what does negative
work mean?

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Well the bicyclist is moving 500
meters in this direction.

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So distance is 500 meters.

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But friction isn't acting
along the same

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direction as distance.

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The whole time, friction is
acting against the distance.

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So when the force is going in
the opposite direction as the

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distance, your work
is negative.

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So another way of thinking of
this problem is energy initial

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is equal to, or you could say
the energy initial plus the

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negative work of friction,
right?

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If we say that this is a
negative quantity, then this

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is equal to the final energy.

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And here, I took the friction
and put it on the other side

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because I said this is going to
be a negative quantity in

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the system.

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And so you should always just
make sure that if you have

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friction in the system, just as
a reality check, that your

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final energy is less than
your initial energy.

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Our initial energy is, let's
just say 38.5 kilojoules.

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What is the negative work
that friction is doing?

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Well it's 500 meters.

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And the entire 500 meters, it's
always pushing back on

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the rider with a force
of 60 newtons.

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So force times distant.

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So it's minus 60 newtons,
cause it's going in the

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opposite direction of the
motion, times 500.

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And this is going to equal
the ending, oh, no.

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This is going to equal the
final energy, right?

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And what is this?

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60 times 500, that's 3,000.

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No, 30,000, right.

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So let's subtract 30,000
from 38,500.

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So let's see.

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Minus 30.

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I didn't have to do that.

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I could have done
that in my head.

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So that gives us 8,455 joules is
equal to the final energy.

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And what is all the
final energy?

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Well by this time, the rider's
gotten back to, I guess we

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could call the sea level.

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So all of the energy
is now going to be

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kinetic energy, right?

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What's the formula for
kinetic energy?

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It's 1/2 mv squared.

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And we know what m is.

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The mass of the rider is 90.

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So we have this is 90.

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So if we divide both sides.

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So the 1/2 times 90.

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That's 45.

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So 8,455 divided by 45.

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So we get v squared
is equal to 187.9.

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And let's take the square root
of that and we get the

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velocity, 13.7.

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So if we take the square root of
both sides of this, so the

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final velocity is 13.7.

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I know you can't read that.

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13.7 meters per second.

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And this was a slightly more
interesting problem because

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here we had the energy wasn't
completely conserved.

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Some of the energy, you could
say, was eaten by friction.

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And actually that energy
just didn't

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disappear into a vacuum.

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It was actually generated
into heat.

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And it makes sense.

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If you slid down a slide of
sandpaper, your pants would

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feel very warm by the time you
got to the bottom of that.

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But the friction of this, they
weren't specific on where the

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friction came from, but it
could have come from the

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gearing within the bike.

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It could have come
from the wind.

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Maybe the bike actually
skidded a little

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bit on the way down.

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I don't know.

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But hopefully you found that
a little bit interesting.

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And now you can not only work
with conservation of

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mechanical energy, but you can
work problems where there's a

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little bit of friction
involved as well.

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Anyway, I'll see you
in the next video.

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