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The centre-of-momentum frame is an inertial frame in which the system's total momentum is zero. For nonrelativistic particles with total mass $M_T$ and total momentum $\mathbf{P}$, it moves with velocity $\mathbf{V}=\mathbf{P}/M_T$ relative to the original frame.

Under a Galilean boost by a constant velocity $\mathbf{U}$, the total kinetic energy transforms asFor an isolated collision, momentum conservation gives $\Delta \mathbf{P}=0$, and nonrelativistic mass conservation gives $\Delta M_T=0$. ThereforeAn elastic collision with $\Delta K=0$ in one inertial frame has $\Delta K^{\prime }=0$ in every inertial frame. Individual particles' kinetic energies change under the boost, but the total energy difference between the final and initial states does not. This is the Galilean invariance of collision elasticity.