Model trajectories for a spinning tennis ball: I. The service stroke

In previous papers in this journal we presented the equations of motion for a rotating spherical projectile, with the spin-vector pointing in an arbitrary direction, and in the presence of an arbitrary wind. Forces included were gravity, the drag force and the lift (or Magnus) force. Most recently we have also included the two inertial forces present on the rotating Earth, the Coriolis and centrifugal forces, and also a ground friction force to allow for a projectile rolling across the ground in sports such as lawn bowls. The equations were solved numerically and used to calculate the trajectories of cricket balls primarily, and to a lesser extent golf balls. In this paper we use the same equations of motion to calculate the trajectories of tennis balls in flight, specifically those of the service stroke, perhaps the most important stroke in modern tennis. We consider the 'flat' serve (a serve delivered without spin), the sliced or side-spin serve, the topspin ('kicking') serve, 'axial-spin' serves where the spin-vector is directed in the same direction as the initial velocity vector, and the more general cases of serves delivered with combinations of these spin types. Variation of the amount of spin, the speed of the serve, the height of the point of delivery and whether the serve is a wide serve or a centre-line serve are considered. Quantitative results for the amount of movement due to spin are included. For example, it is found that, for a spin of 600 rad s(-1) (5730 rpm), a sliced serve can move side-ways by similar to 2.65 m at the point of impact with the ground, a top-spinning serve may impact the ground similar to 9.5 m closer to the server than an otherwise identical flat serve, and an axial-spin serve may move side-ways by similar to 40 cm.