Non-smooth modelling of billiard- and superbilliard-ball collisions

A description of billiard-ball collisions using a ‘discontinuous’ model is presented considering a two-step situation corresponding to the ball–ball interaction followed by ball-supporting surface interaction. It is applied to the inelastic impact of a cue ball having arbitrary pivotment and ‘English’ spins against an object ball initially at rest. This formulation provides a simplified approximation to the ‘continuous’ models of impact and considers two different regimes of impact: gross slip, and slip–stick, described in terms of coefficients of friction and restitution. As a result, the angles of scattering of the balls just after the impact (post-collision angles) and when the ball reach pure rolling motion (post-transition angles) are expressed in terms of the angle of impact, the mass ratio, and the initial spin conditions. Theoretical predictions are compared with experimental data for different materials, including regulation billiard ball and superballs.