Radar speed gun true velocity measurements of sports-balls in flight: application to tennis

Spectators of ball-games often seem to be fascinated by the speed of delivery of the ball. They appear to be less interested in or even oblivious to the mechanism and accuracy of the measurement or where in the flight path of the ball the measurement is actually made. Radar speed guns using the Doppler effect are often employed for such speed measurements. It is well known that such guns virtually always measure the line-of-sight or radial velocity of the ball and as such will return a reading less than or equal to the true speed of the ball. In this paper, using only basic physics principles we investigate such measurements, in particular those associated with the service stroke in tennis. For the service trajectories employed here, a single radar gun located in line with the centre-line of the court in fact under-estimates the speed of a wide serve by about 3.4% at the point of delivery, and by about 14.3% on impact with the court. However, we demonstrate that both the magnitude and direction of the true velocity of the ball throughout its entire flight path may be obtained, at least in principle, by the use of four suitably placed radar speed guns. These four guns must be able to measure the 'range' to the ball, enabling its position in flight to be determined, and three of them must be able to measure the radial velocity of the ball. Restrictions on the locations of the speed guns are discussed. Such restrictions are quite liberal, although there are certain configurations of the radar gun positions which cannot be used. Importantly, with the one proviso that no speed gun can be directly in the path of the ball (not only for the obvious reasons), we find that if the speed of the ball can be determined for one point in the trajectory, it can also be determined for all points. The accuracy of the range and radial velocity measurements required to give meaningful results for the true velocity are also briefly discussed. It is found that the accuracy required for these measurements depends on the location of both the radar guns and the ball itself, and must be determined for each particular situation.