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Let's examine when the momentum
of a system is constant,

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and apply that to
solving problems.

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First, we'll revisit
the impulse equation.

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On the left, we
have the impulse,

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the integral of the total
external force acting

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on a system between some
initial and final times.

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And on the right,
we have the change

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in the total momentum
of the system,

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between those initial
and final times.

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In a situation where
the impulse is 0,

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we have that the change
in the momentum is 0.

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P initial is equal to p final.

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In other words, the
momentum of the system

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is constant between some
initial and final time.

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This is a vector equation.

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But in many problems, we can
set up our coordinate system

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so we only have to
consider one dimension.

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Here's a simple example
of one block moving along

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a horizontal surface with
some initial velocity,

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colliding with a block at rest.

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And then the two blocks
stick together and move.

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Is momentum constant
during this collision?

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To answer that question,
we need to think

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about what our system is.

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Let's look at the system
of both blocks together.

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We can see the external forces,
gravity and the normal force,

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add up to 0.

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So our total
external force is 0.

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During the collision, each block
exerts a force on the other.

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But that is an internal
force if both blocks

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are included in our system.

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You can see that if our system
was just one of the two blocks,

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we would need to know the
collision force in order

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to solve for the final speed.

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Therefore, we will
choose our system

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to be the two blocks together.

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We then need to set up
a coordinate system.

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We'll pick the origin here, and
this direction to be positive

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x.

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Now we need to identify the
initial and final states.

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Our initial state will be just
before the blocks collide.

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The momentum of the
system is the sum

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of the momentum of each
block individually.

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We have m1 times v1 initial
x times i-hat plus m2 times

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v2 initial x times i-hat.

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This second term is 0, since
block 2 starts at rest.

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The final state is right
after the collision

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when the two blocks
are moving together.

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They have the same velocity,
so the final momentum

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is m1 plus m2 times v
final x times i-hat.

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This gives us an
equation that will

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allow us to solve for whatever
quantity we are not given.

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Now let's look at a similar
collision, one that's

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happening in two dimensions.

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If we have one block
coming up from the bottom,

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hitting one that's
coming in from the side,

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I'll choose my unit vectors to
be like this, my origin here.

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The momentum equation
gives us two equations,

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one along the x
direction and the other

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along the y direction.

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Once again I'll choose
the initial state

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to be just before the
collision and the final state

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to be just after the collision.

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The momentum in
the initial state

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is again the sum of the momentum
of each block individually.

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And the momentum
in the final state

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is the mass of the two
blocks added together

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times the velocity
of the two blocks.

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Notice that this velocity
has a component in both

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the x and the y directions.

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Again the total external
force on the system is 0.

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So the impulse is 0 in both
the x and y directions,

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and therefore, momentum is
conserved for both the x and y

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components.

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So I can write my two
momentum equations like this.

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The impulse equation
is a vector equation.

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So generically,
in two dimensions,

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we will have two equations.

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We always need to check that
the momentum is conserved

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in each direction separately.

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And it is possible to
have a case where momentum

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is conserved in one
direction but not the other,

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if we have some net external
force in one direction.

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What would happen if, in
our 1d collision example,

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the blocks now experienced
friction along the surface?

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Can we still assume that
the momentum is conserved?

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In fact, if we pick a point in
time right before the collision

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and compare that to a point in
time right after the collision,

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we can see that the
impulse from friction

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is over such a short
period of time,

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that the impulse
is really small.

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And we can say that the momentum
is approximately constant.

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If we consider times later,
after the collision is over,

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then momentum is
certainly not conserved,

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which is what you would expect.

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You'll see the two blocks
slow down due to friction.

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So even if there are other
forces acting on the system,

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like friction or
gravity, we can still

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calculate the result
of a collision

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as if the momentum is
constant during the collision

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by picking times only a
very small delta t apart.