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So this is the initial momentum, that's going to be the final momentum.
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That also means initial momentum square is final momentum squared.
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If two vectors are equal, the squares are equal.
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Now, the final momentum squared,
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I say we can evaluate not in the lab frame,
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but in another frame called the center of mass frame.
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And here is a center of mass frame.
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Suppose you start moving to the right at some speed,
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what do you think will happen?
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Imagine in your head.
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You move to the right, this proton will start moving to the left,
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this proton will appear slower.