4.2.8 high speed collisions

Here we discuss uses for the relative proper-velocity addition rule to examine collisions at any speed, from more than one point of view. 

This starts with the observation that momentum conservation (pAC+pBC = p'AC+p'BC) with respect to frame C may be described in terms of individual-object momentum changes before and after (i.e. where Δ means "final minus initial") as:

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Using the proper-velocity addition equation from the section in "adding vector velocities", in frame D's perspective (with respect to which frame C moves with proper velocity wCD), only two terms survive on each side of the equation since the ΔwCD factor in the third (clock-change) term on both sides of the equality is zero. As a result we can say quite generally that frame D's "momentum-difference equation" simply scales components of frame C's equation differently, according to their alignment with wCD i.e. the parallel component is multiplied by γCD ≡ Sqrt[1+(wCD/c)2]:

. 

Hence if the sum of momenta before and after collision is constant in one frame, it's constant in all.