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When we analyzed a
one-dimensional elastic

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collision in any
frame and we had

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a velocity V1 initial
and V2 initial,

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that we saw that
we could reduce--

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this is a one-dimensional
elastic collision--

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we always have the energy
condition and the momentum

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condition.

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But when we combine the
energy movement together,

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we found the following idea
that the relative velocity

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V1, 2, which we called the
relative velocity initial,

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was just equal to the
final relative velocity.

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The statement there was
that V1 initial minus V2

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initial is equal to minus
V1 final minus V2 final.

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And we call this the
energy momentum relation

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for our classical mechanics.

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And then we could combine that
with our conservation of energy

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law, conservation of momentum.

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And we've got a linear
system of equations

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that is much easier to solve
than the quadratic system.

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Now what I'd like to
show is that this concept

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of relative velocity, that
V1, 2 relative velocity

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is independent of the
choice of reference frame.

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And so if we want to
analyze a collision

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in any other reference
frame, then we always

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can keep this result. So
now let's look at that.

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So again, let's imagine that
we have two particles, where we

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have particle 1 and particle 2.

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We have some origin, r1 and r2.

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And now we want to choose
another reference frame.

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So suppose that we pick a second
reference frame, which has

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maybe some origin over here.

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And we have the vectors
r1 prime and r2 prime.

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And the relative vector from
the center of one reference

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frame to the center of
the other reference frame.

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So what we have are
the two conditions

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that r1 is equal to
capital R plus r1 prime.

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r1 is capital R plus r1 prime.

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r2 is capital R plus r2 prime.

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And now if we subtract
these two equations,

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we have r1 minus r2 equals
r1 prime minus r2 prime.

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And this shows us that
the relative position

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vectors-- and let's draw
that this one from 2 to 1.

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And even in our diagram, we can
see that the vector from 2 to 1

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does not depend on the
choice of reference frame.

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And even more importantly,
when we differentiate,

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we get that V1 minus V2 is equal
to V1 prime minus V2 prime.

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And that becomes our statement
that the relative velocity

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vector is independent of the
choice of reference frame.