A unique pure mechanical system revealing dipole repulsion

TL;DR

This paper analyzes a one-dimensional elastic collision system revealing a dipole-like repulsive force, with an exact analytic solution for the block's trajectory and a force proportional to 1/x^3 near the turning point.

Contribution

It introduces a unique mechanical system demonstrating dipole-like repulsion through elastic collisions and provides an exact analytic solution for the block's motion.

Findings

Force near the turning point is proportional to 1/x^3

Analytic expression for the block's trajectory

Reveals dipole-like repulsion in a pure mechanical system

Abstract

We study multiple elastic collisions of a block and a ball against a rigid wall in one dimension. The complete trajectory of the block is solved as an analytic function of time. Near the turning point of the block the force carried by the ball is proportional to 1/x^3, where x is the distance between the wall and the block, in the limit that the block is sufficiently heavier than the ball. This is a unique pure mechanical system that reveals dipole-like repulsion.