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We would like to now apply
the momentum principle

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to examples of recoil.

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So recall that the
momentum principle

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is that the external
force causes the momentum

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of the system to change.

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Now, this is a vector equation.

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So for example, if the external
force in the x direction

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is 0, then the momentum of
the system in the x direction,

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let's say, the final
momentum will be

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equal to the initial momentum.

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I would like to now apply
the momentum principle

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to an example of recoil.

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In our recoil example, we have
a person jumping off a cart.

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So let's just look at
how this example can work

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if we draw momentum diagrams.

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So suppose we choose
a ground frame.

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And in that ground frame,
we have a cart and a person.

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A person is standing
on the cart.

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And this is t initial.

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And here, they are at rest.

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Now, the person we're going
to assume to jump horizontally

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off the cart, and
the cart will recoil

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in the opposite direction.

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So after the jump, we can
describe this picture.

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The person is moving
with the velocity,

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v. The cart is moving with vc.

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And the person has jumped
with the velocity vp.

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Now, suppose I choose a
different reference frame.

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That instead of
choosing a ground

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frame, as a reference frame
moving with the velocity vc.

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You can imagine that
maybe I have a car here

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and you're inside that car
moving with velocity vc,

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and you're looking
at this picture.

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Then, what would our
momentum diagrams look like?

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Well, if I'm moving
in a car this way

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and in the ground frame
the initial picture

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is the person in
the cart is at rest.

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Then, in my moving
frame, it actually

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looks as if the
cart and the person

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are moving in the
opposite direction.

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So let's write that this way.

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Here's the initial
picture, t initial.

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And in this frame, the
person and the cart

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are moving with vc minus vc.

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I put an arrow here
to indicate that it's

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opposite the
direction of vc there.

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But their velocity
is minus the velocity

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of the reference frame.

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After the jump-- so
here's the person now.

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The cart is at rest, why?

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Because we're in the reference
frame moving with vc.

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So if you're in a car and
you're moving at the same speed

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that the cart has
with the ground frame,

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then in your frame, this
cart looks like it's at rest.

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What about the
person jumping off?

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Well, let's write it this way.

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So this is the velocity.

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I'm going to use a symbol, u.

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Now, u-- this is what
do we mean by that.

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This is the velocity of
the person in the moving

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frame that's moving with
velocity vc-- that's

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the velocity of the
person as seen by a car.

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Sometimes we call this
the velocity of the person

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relative to the cart.

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What does that word
relative to the cart mean?

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Well, you can see
in this picture.

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In this moving frame,
the cart is at rest,

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and the person jumps
with the speed u velocity

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u relative to the cart.

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So these are momentum
diagrams for a ground frame

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in which the person
in the cart started

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at rest, the person jumps off.

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I can put an arrow
here, but it's really

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information is in that vector.

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The cart is moving.

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In a frame moving with
the velocity of the cart,

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then what does my
picture looks like?

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Well, the cart and
the person initially

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are moving opposite directions.

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Again, you're moving this way.

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The cart looks like
it's moving that way

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if you're inside the car.

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And the final state,
person, cart is at rest

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and the person is jumping
with the velocity u

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relative to the cart.

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Now, our question now
is how do we relate

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these two velocities, u and vp?

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What are u and vp?

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vp is the velocity of the
person in the ground frame.

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And u is the velocity of the
person in the moving frame.

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Well, we've already
seen our equation

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for how to get velocities
in different frames.

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We have that vp equals the
relative velocity of the two

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frames plus the velocity
in the moving frame.

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So what we have is, this is
the velocity of the person--

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let me just clean that up-- of
the person in the ground frame.

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v is the relative velocity
of the two frames.

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So here we have that v is
the velocity of the cart,

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because you're in a frame
moving with vc with respect

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to the ground.

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And u is the velocity of the
person in the moving frame.

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So this is how we can show
the same type of interaction

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in two different
reference frames.

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Next, we'll figure out
what these velocities are.