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

Abstract

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 V-n whose components are proportional to the velocities of the objects. Energy-momentum conservation then requires a covariant recurrence relation for 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." (C) 2015 American Association of Physics Teachers.