WEBVTT

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Let's consider a
ball that is dropped

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from a certain height, h i,
above the ground and this ball

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is falling.

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It hits the ground
and it bounces up

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until it reaches some
final height, h final.

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Now when the ball is
colliding with the ground,

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there are collision forces.

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And in this problem
what we like to do

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is figure out what the
average force of the ground

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is on the ball.

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And that will be
the normal force,

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the average normal force, on
the ball during the collision.

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Now if we look at
this ball dropping,

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it's going to lose a
little bit of energy,

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because it's getting
compressed at the collision.

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Let's look at an example of
the actual ball dropping.

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As you can see in
this high speed video,

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as the ball falls down, it
collides with the ground.

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When it collides with the
ground, it's compressed.

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And then as it rebounds
upwards, the ball

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expands back to
its original shape,

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but it doesn't quite get
to the same height-- that's

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because when the
ball is compressed,

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there's some deformation in the
rubber structure of the ball,

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and it's not a completely
elastic deformation.

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And so some of the energy
is transformed into, first,

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molecular motions, which turn
into thermal energy that's

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radiative into the environment.

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Let's look in particular at
the details of the collision.

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If we look at it in slow
motion what we have here--

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and I'll draw a picture--
as the ball is colliding

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with the ground,
ball compresses,

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expands as it goes upwards.

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And so we can draw a free
body diagram of the ball

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with a normal force and
a gravitational force.

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Now let's choose our
positive direction up.

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So now what we'd like to do is
apply the momentum principle

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to analyze the
average normal force.

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And our momentum principle,
remember is impulse.

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The force integrated over
some time during the collision

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is equal to the
change in momentum.

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So what we'd like
to do is identify

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the states that are relevant.

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So it will have a state
before, so what we'll do

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is we'll call this
the before state,

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and that's right before the
ball is hitting the ground.

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And we have an after state,
and in the after state,

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the ball has now finished
colliding with the ground,

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and it's now moving
up with speed up.

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Now again, we're going
to choose a positive up.

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Here on representing
things as it speeds.

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One of the things, we
need some times here,

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so let's say that t initial is
zero, this is our final time.

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We'll call this time
the before time,

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we'll just call this t
before, and this is t after.

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And then our integral
is going from before to

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after the momentum.

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And we can now apply
the momentum principle.

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Well, this is a vector equation
and we've chosen unit vectors

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up, so what we have
here is the integral

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of from t before to t after of
N minus mg, integrated over dt,

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and that's equal to
the momentum at the y

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component of the
momentum at t after,

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minus the y component
of the momentum.

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We don't have a
vector here anymore.

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The y component of
the vector, t before.

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And so this is our expression
of the momentum principle.

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Impulse causes
momentum to change.

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Now we're assuming that the
normal force just averaging it

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and so this intregral simply
becomes N average minus

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ng, times the time of collision,
is equal to-- now in here

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we can put the mass of the
ball, we have the velocity.

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Now here's where we
have to be a little bit

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careful, because we're
looking at the y component.

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We chose speed downwards, that's
in the negative y direction,

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so we have minus-- sorry,
we're looking at after.

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We have plus V after, because
this is going in the positive j

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direction.

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And over here we
have a negative mass,

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but it's going in
the minus direction,

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so we have a minus
mV before, and so we

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get mass times V
after plus V before.

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So our first result is that
the normal force average.

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Let's bring the divide
through by delta t,

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and bring the Ng term over,
so we have m Va plus Vb,

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divided by delta t, plus mg.

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So we see that if the
collision time is very short,

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then this average force
is a little bit bigger.

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A long collision time,
the average force

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a little bit smaller.

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Now from kinematics, we already
have worked out the problem

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that the speed
for an object that

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rises to a height, h final,
this is the velocity afterwards,

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is just square
root of 2g h final.

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And in a similar way, if an
object is falling height h i,

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the speed when it gets
to the bottom is 2ghi.

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And so now we can conclude
with these substitutions

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that the average force equals
m times square root of 2g h

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final, plus the square
root of 2g h initial,

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over the collision
time, plus and mg.

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And of course the
collision time,

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we're saying is t
after minus t before.

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And so that's how we can
use the momentum principle

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to get an average expression
for the normal force.