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We would like to now introduce
a new methodological tool

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for analyzing problems that
involve momentum transfers.

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And we call that tool
momentum diagrams.

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Now, what we'd
like to do is look

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at our fundamental idea, which
was, for discrete objects,

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we have that-- involved
in a collision-- we have

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that the external force
integrated with respect

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to time-- in poles-- is equal
to the change in momentum

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between two different states.

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So this is the momentum
of the final state,

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and this is the momentum
of the initial state.

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So when we want to
analyze problems,

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how can we methodologically
introduce a picture

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representation of our problems?

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So let's look at what
we need to do first.

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We always need to choose a
system that we're referring to.

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And with respect to the
choice of that system,

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we need to also choose
a reference frame.

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Now, once we've done
that, we can now

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represent a collision with
respect to this system.

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For instance, let's consider
two objects-- as our system

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Object 1 and Object 2.

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They're moving on a
frictionless, horizontal

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surface.

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And we're choosing as a
reference frame, the ground

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frame.

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Now, what we'd like to do
is identify two states.

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So for our initial
state, we'll have--

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if we're given some
initial conditions-- what

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we'd like to do is represent
each object with a velocity

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vector.

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So here, we'll
write the 1 initial,

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and let's suppose this object
is coming at it with the 2

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initial as an example.

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And once we've represented
the velocities of the objects,

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we can write down their
momentum in the initial state.

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Similarly, if these two objects
collide and they're moving,

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we actually don't know which
way those objects will end up

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moving.

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And so what we'd again like
to do in our final state,

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after this collision, is to
represent the velocities by,

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again, vectors.

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So this is V1 final
and this is V2 final.

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And then, we can represent
the change in momentum.

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So our momentum principle
now becomes-- in this case,

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let's just assume that
the external force here

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sums to zero-- we're
assuming no friction.

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And then our momentum
principle says that 0

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equals P final minus P initial.

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And now, we can read
off those momentums

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as vectors on the diagram,
and so what we have

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is that the final
momentum will be

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equal to the initial momentum.

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So we can write down
M1 final plus M2

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V2 final is equal to M1 V1
initial plus M2 V2 initial.

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And that's how we can represent
a collision where there

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is no external forces and
use our momentum principle

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to get an vector equation.

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Now, in many problems,
you're given information.

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You might be given information
about the speeds and magnitudes

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of the objects.

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And in order to then
take this equation

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and represent it in speeds and
directions or even components,

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we need to choose some
coordinate system.

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So if we choose a coordinate
system-- and that's

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the third step-- so suppose
we choose a coordinate system,

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then we can start to look at
two different representations

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for our problem.

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For instance, let's just choose
this to be the i hat direction.

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Now, given that choice, we
could describe the velocities

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in terms of components.

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Now this gets awkward--
V1i x component i hat.

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And similarly, we can write
down all the velocities-- V2i--

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as V2ix i hat, et cetera,
for all the velocities.

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And our momentum
equation in components

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then becomes, in the i
hat direction, M1 V1ix

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plus-- well, here we
have the final state,

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so let's make this consistent--
the final plus M2 V2 final x

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equals M1 V1 initial x
plus M2 V2 initial x.

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And that's the same equation
that we have as vectors, now

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expressed in terms
of components.

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And recall that components
can be positive or negative.