Misconception Concerning the Dynamics of the Impact Ball Apparatus

Abstract

The impact ball apparatus, which usually consists of a straight row of balls of equal size touching each other, is used for demonstrating conservation of momentum. If the balls are perfectly elastic and there are more than two balls in the row, then if N balls are drawn aside and allowed to strike the remaining group, it is-just not true, as is often erroneously supposed, that N balls fly out from the other end of the row while the other balls remain at rest. The two principles of conservation of momentum and of energy are sufficient to determine the motion after impact of only two balls. If there are more than two, then the law of force between the balls (a 32 power law) must be invoked. The case for three equal masses with linear force laws is solved exactly and several consequences are discussed. Numerical solutions for the cases of three and four equal spheres are given. Calculations are found to agree with experimental data. When a single ball of velocity v0 impacts against two balls in contact at rest, the velocities after impact are −0.0710v0, +0.0765v0, and +0.9944v0, if the balls are elastic. Since the demonstration is always interesting and has many ramifications, it seems unnecessary to mislead students into believing that the velocities of the balls with relatively little motion arise because of inelasticity or misalignment, neither of which can explain the negative velocity. If the coefficient of restitution is much less than unity, then inelasticity masks the effects discussed here.