A-Level Impulse and the conservation of momentum and energy: worked solution

Question

The figure shows two rigid spheres, P and Q, each of mass m, moving towards one another with speeds u_1 and u_2 respectively as measured in the laboratory frame. Demonstrate that, in the centre-of-mass frame, the combined momentum of the two spheres is zero just before they collide.

Take P moving in the +x direction with speed u_1 and Q moving in the -x direction with speed u_2 (they approach each other). Total lab momentum = m*u_1 - m*u_2. The centre-of-mass velocity is V = (m*u_1 - m*u_2)/(m + m) = (u_1 - u_2)/2. In the CM frame each velocity is reduced by V, so the total momentum is: m(u_1 - V) + m(-u_2 - V) = (m*u_1 - m*u_2) - 2mV = (m*u_1 - m*u_2) - 2m*(u_1 - u_2)/2 = (m*u_1 - m*u_2) - (m*u_1 - m*u_2) = 0. Hence the total momentum in the CM frame is zero.