To produce new particles, accelerators often smash two equal-mass objects together—proton and proton or electron and positron. The threshold energy is the kinetic energy before the collision needed simply to produce the final particles—their mass/internal energy alone, with no leftover kinetic energy. Consider a colliding-beam accelerator in which two initial particles of mass m are moving at the same speed relative to the lab. Assume that the total mass of the (stationary) particles after the collision is M. Show that the threshold energy is (M - 2m)c2.
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