= Solution
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 as
$$
K'=\sum_i\frac{m_i}{2}|\mathbf v_i-\mathbf U|^2=K-\mathbf U\cdot\mathbf P+\frac12M_TU^2.
$$
For an isolated collision, <momentum conservation> gives $\Delta\mathbf P=0$, and nonrelativistic <mass> conservation gives $\Delta M_T=0$. Therefore
$$
\boxed{\Delta K'=\Delta K.}
$$
An <elastic collision> with $\Delta K=0$ in one <inertial frame> has $\Delta K'=0$ 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>.
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