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COM frame in Relativistic particle physics

Discussion in 'Physics' started by Universe Maintainer, Oct 8, 2018.

  1. I have a home work:

    Consider a head-on collision of a 'bullet' of rest mass $M$ with a stationary 'target' of rest mass $m$. Prove that the post-collision γ-factor of the bullet cannot exceed:$$(m^2 + M^2)/(2mM).$$ [Hint: if $P, P'$ are the pre- and post- collision four-momenta of the bullet, and $Q, Q'$ those of the target,show, by going to the CM frame, that $(P' − Q)^2 ≥ 0$; in fact, in the CM frame $P' − Q$ has no spatial components.]

    I have solved problem from the hint that $(P' − Q )^2 ≥ 0$ and $P' − Q$ should have non-spatial component. The first hint is easy, because four-momentum is time-like hence it should always be non-negative. What i don't understand is that why $P' − Q$ should have non-spatial component in COM?

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