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Let's consider our
one-dimensional collision

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again, object 1 moving
with velocity V1 initial,

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and object 2 moving
with V2 initial.

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Let's call this or
i hat direction.

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And this is our initial state.

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And our final state after the
collision, we have object 1.

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We'll say it's moving this,
V1 final, and object 2 moving

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that way, V2 final.

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Now recall our principle
of impulse and momentum.

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We said that if there
is an external force

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during the time of a collision
delta t, then physically that

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will cause the momentum
of the system final

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minus the momentum of
the system initial.

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Now when we do this analysis,
this side was a description

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and this side is physics.

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Now for our
one-dimensional collision,

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we need to look
at this collision

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and ask ourselves are
there any external forces

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acting on the system, which
is consisting of particle 1

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and particle 2?

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So what we're going
to identify here

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is that the surface
is frictionless.

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And we'll ignore
all air resistance.

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And so by our assumptions that
there are no f external is 0.

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And therefore, the momentum of
the system remains constant.

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So here our statement is-- many
people call this conservation

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of momentum, but we're
saying in this example based

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on our assumptions that
the momentum of the system

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is constant.

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Now how do we actually
write that down?

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Well, let's now write it
first as vector expressions.

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So we have the initial momentum,
V1 of the system, m1 m2 V2

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initial is equal to the
final momentum of the system,

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V1 final plus m2 V2 final.

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Now, how do we represent
these equations?

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Well, you could treat them
as vectors if you wanted.

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But what we're going to do is
express them as components.

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So if we wrote
this as components,

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we would have m1 Vx initial i
hat plus m2 V2 x initial i hat

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equals m1 V1 s final i hat
plus m2 V2 x final i hat.

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So that's the vector
expression expressed

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in terms of components.

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The advantage of this
is that we really

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don't know the signs of
these two final components.

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That's our target quantities.

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But we could just
write this equation--

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instead of writing it
as a vector equation,

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let's just now write this
as a component equation.

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And when we write this equation
in terms of components,

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we have m1 V1 x initial plus
m2 V2 x initial equals m1 V1 x

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final plus m2 V2 x final.

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And this equation
here is the equation

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that we use to
express the constancy

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of the momentum of the system.

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We'll call this equation 1.

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Now our next approach
is to ask are there

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any other quantities in the
system which are constant?