Resolving Kane's Puzzle in Oblique Collisions of Rigid Bodies

TL;DR

This paper provides a new closed-form solution for oblique collisions of rigid bodies, addressing Kane's puzzle by resolving inconsistencies in previous models and ensuring energy conservation without additional parameters.

Contribution

It introduces a nonlinear, energy-conserving solution to oblique collisions that corrects prior misconceptions and resolves Kane's puzzle in classical mechanics.

Findings

Derived a nonlinear relationship between post-collision velocity and initial parameters

Ensured energy conservation without introducing new material parameters

Resolved Kane's puzzle by correcting previous fallacies in collision analysis

Abstract

We examined the asymmetric deformation in collisions and the transition conditions from oblique to normal collisions and non-collisions to address the problem of oblique collisions of rigid bodies in classical mechanics. A closed solution satisfying the fundamental equations and adhering to the energy conservation law without introducing new material parameters was derived. The solution exhibited a nonlinear relationship between post-collision velocity and initial state parameters, contrasting with the linear results of existing studies. This solution avoided the fallacy in Whittaker's hypothesis that kinetic energy might increase after a collision. Consequently, the solution presented herein fundamentally resolves Kane's puzzle, previously overlooked in classical mechanics.