Magic mass ratios of complete energy-momentum transfer in one-dimensional elastic three-body collisions

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 $m=\alpha M$. Initially, one block is incident upon the ball with the other block at rest. For $\alpha<1$, the three objects will make multiple collisions with one another. In our analysis, we construct a Euclidean vector $\textbf{V}_n$ whose components are proportional to the velocities of the objects. Energy-momentum conservation then requires a covariant recurrence relation for $\textbf{V}_n$ that transforms like a pure rotation in three dimensions. The analytic solutions of the terminal velocities result in a remarkable prediction for values of $\alpha$, in cases where the initial energy and momentum of the incident block are completely transferred to the scattered block. We call these values for $\alpha$ "magic mass ratios."