Two-dimensional collisions and conservation of momentum

TL;DR

This paper discusses the analysis of two-dimensional collisions, highlighting the underdetermined nature of the problem and proposing a lab exercise to uniquely determine collision outcomes.

Contribution

It introduces a method for assigning lab exercises that allow students to determine collision outcomes in 2D collisions despite apparent underdetermination.

Findings

A specific class of 2D collisions can be uniquely analyzed.

Educational exercises can help resolve underdetermination in 2D collision analysis.

The approach enhances understanding of momentum conservation in multiple dimensions.

Abstract

Analysis of collisions is standardly included in the introductory physics course. In one dimension (1D), there do not seem to be any unusual issues: Typically, the initial velocities of the two colliding objects are specified, and the problem is to find the final velocities. In 1D there are therefore two unknown variables. One can write the equation for conservation of momentum, and either the equation for conservation of energy for the perfectly elastic case, or the expression for the coefficient of restitution (COR) otherwise. Thus, one has two equations for two unknowns, and one may solve the problem fully. An issue arises, however, in two-dimensional (2D) collisions: There are four unknown variables (two components of the final velocity of each object), but now there appear to be only three equations: two components of the equation of conservation of momentum, and the energy condition. The problem may appear therefore to be underdetermined. If this problem were in principle an underdetermined one, one would fail in predicting the outcome of the collision experiment. We describe how one may assign students an appropriate lab exercise and problems for an interesting class of 2D collisions for which one can determine uniquely the outcome of the collision.