Show that for relativistic, elastic scattering of two particles in the COM frame, that each particle has the same kinetic energy after scattering as before. That is, each particle can have different direction for its velocity after scattering, but it must have the same magnitude for its velocity. Hint: Use E12−p12=m12=E1′2−p1′2, a similar relation for particle \#2, and the 3-momentum balance equation [where total 3-momentum is zero in the COM] to find E12−E1′2=E22−E2′2. Divide this by the energy balance equation. Then add and subtract the energy balance equation from your result.