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 ${\text{V}}_n$ whose components are proportional to the velocities of the objects. Energy-momentum conservation then requires a covariant recurrence relation for ${\text{V}}_n$ 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."

32 pages, 6 figures, 2 tables

References in corpus (3)

• The Hertz contact in chain elastic collisions
• A Billiard-Theoretic Approach to Elementary 1d Elastic Collisions
• A unique pure mechanical system revealing dipole repulsion

Cited by in corpus (1)

• Delayed Rebounds in the Two-Ball Bounce Problem