m 1 u 1 1 − − u 1 2 / c 2 + m 2 u 2 1 − − u 2 2 / c 2 = m 1 v 1 1 − − v 1 2 / c 2 + m 2 v 2 1 − − v 2 2 / c 2 = p T m 1 c 2 1 − − u 1 2 / c 2 + m 2 c 2 1 − − u 2 2 / c 2 = m 1 c 2 1 − − v 1 2 / c 2 + m 2 c 2 1 − − v 2 2 / c 2 = E We can look at the two moving bodies as one system of which the total momentum is p T , the total energy is E and its velocity v c is the velocity of its center of mass. Relative to the center of momentum frame the total momentum equals zero. It can be shown that v c is given by: v c = p T c 2 E Now the velocities before the collision in the center of momentum frame u 1 ′ and u 2 ′ are: u 1 ′ = u 1 − − v c 1 − − u 1 v c c 2 u 2 ′ = u 2 − − v c 1 − − u 2 v c c 2 v 1 ′ = − − u 1 ′ v 2 ′ = − − u 2 ′ v 1 = v 1 ′ + v c 1 + v 1 ′ v c c 2 v 2 = v 2 ′ + v c 1 + v 2 ′ v c c 2