We consider the one-dimensional scattering of two identical blocks of mass M that exchange energy and momentum via elastic collisions with an intermediary ball of mass . Initially, one block is incident upon the ball with the other block at rest. For , the three objects will make multiple collisions with one another. In our analysis, we construct a Euclidean vector whose components are proportional to the velocities of the objects. Energy-momentum conservation then requires a covariant recurrence relation for that transforms like a pure rotation in three dimensions. The analytic solutions of the terminal velocities result in a remarkable prediction for values of α, in cases where the initial energy and momentum of the incident block are completely transferred to the scattered block. We call these values for α “magic mass ratios.”
Let us denote by ui and vi the initial and final velocities of the ith object, respectively. Then conservation of energy and momentum require and . These coupled equations have a unique set of solutions: v1 = u2 and v2 = u1. If they do not make a collision, then v1 = u1 and v2 = u2.
Let us consider the last collision between C and B at Qk when . At Qk the velocity of C changes from v to 0 and that of B changes from to V. Energy-momentum conservation requires and . The solutions to v and are and . In a similar manner, we find that the velocities of A and C must be and , respectively, just before the collision at Pk. This guarantees that A stops at Pk and C moves with the velocity between Pk and Qk.
Note that α = 0 represents the case that the ball is absent.