The motion of an arbitrarily rotating spherical projectile and its application to ball games

Abstract

In this paper the differential equations which govern the motion of a spherical projectile rotating about an arbitrary axis in the presence of an arbitrary 'wind' are developed. Three forces are assumed to act on the projectile: (i) gravity, (ii) a drag force proportional to the square of the projectile's velocity and in the opposite direction to this velocity and (iii) a lift or 'Magnus' force also assumed to be proportional to the square of the projectile's velocity and in a direction perpendicular to both this velocity and the angular velocity vector of the projectile. The problem has been coded in Matlab and some illustrative model trajectories are presented for 'ball-games', specifically golf and cricket, although the equations could equally well be applied to other ball-games such as tennis, soccer or baseball.