Galilean invariance of collision elasticity ($\Delta K^{\prime }=\Delta K$)

For fixed particle masses, a Galilean boost by $\mathbf{U}$ changes total kinetic energy to $K^{\prime }=K-\mathbf{U}\cdot \mathbf{P}+M_T{U}^2/2$. An isolated collision conserves total momentum $\mathbf{P}$ and total nonrelativistic mass $M_T$, so its final-minus-initial kinetic energy change is independent of the inertial frame. An elastic collision in one such frame is elastic in all of them. Conservation of momentum, rather than invariance of the numerical value of K, is essential to the argument.