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Numerical and analytical calculations are presented on the flight of a tennis ball and the subsequent bounce off the court for typical groundstrokes. The horizontal flight distance was chosen as 22.77 m, corresponding to a typical shot launched from above the baseline to land 1 m inside the opposite baseline. A wide range of shots was examined by varying the launch speed, launch angle and topspin, using representative values of the lift and drag coefficients, as well as representative values of the coefficients of restitution and friction to determine the launch parameters after the bounce. The results compare favourably with measurements gathered from Hawk-Eye data, and could be used to analyse trajectory data in greater detail and to estimate the effectiveness of different types of groundstrokes. [ABSTRACT FROM AUTHOR]

Measurements and calculations are presented on the vibration of the handle of a tennis racquet. It is shown that the vibration amplitude depends primarily on the stiffness of the racquet, but it also depends on the mass of the racquet, the mass distribution, the vibration frequency, the impact location, the impact duration, the stiffness of the ball and the stiffness of the string plane. Mass added at the tip of a racquet usually reduces handle vibrations when the ball impacts near the tip, but it increases handle vibrations for impacts in the throat region. Mass added to the handle usually reduces handle vibrations, regardless of the impact point. Mass added at a vibration node point has no effect on the vibration frequency, only a small effect on amplitude, and can usefully be added to very light and stiff racquets to increase racquet power without affecting the low vibration amplitude of these racquets. The amplitude of the fundamental vibration mode decreases as the impact duration increases, decreasing to zero at all impact points when the impact duration is 1.5 times longer that the fundamental vibration period. [ABSTRACT FROM AUTHOR]

The Hawk-Eye line-calling system in tennis uses a footprint of the ball to determine whether the ball was in or out. However, the footprint itself is not measured. It is the ball trajectory through the air that is measured. Measurements of the footprint are presented in this paper, showing how the high initial rate of expansion of the footprint and other technical difficulties can lead to potential errors in the Hawk-Eye system. The footprint was measured for vertical and oblique impacts, and is compared with recent results presented at a tournament where Hawk-Eye was installed. [ABSTRACT FROM AUTHOR]

Measurements are presented of drag and lift on new tennis balls in flight. Two video cameras were used to measure the velocity and height of the balls at two positions separated horizontally by 6.4 m. The balls were fired from a ball launcher at speeds between 15 and 30 m/s and with topspin or backspin at rates up to 2,500 rpm. Significant shot-to-shot variations were found in both the drag and lift coefficients. The average drag coefficient was 0.507 ± 0.024, independent of ball speed or spin, and lower than the value usually observed in wind tunnel experiments. The lift coefficient increased with ball spin, on average, but significant lift was observed even at very low spin. The latter effect can be attributed to a side force arising from asymmetries in the ball surface, analogous to the side force responsible for the erratic path of a knuckleball in baseball. [ABSTRACT FROM AUTHOR]

A common problem in sports mechanics is to understand and to model the impact of a sports ball with some other object such as a sports surface or an implement used to strike the ball. In this review, we focus on a simple collision model that effectively reduces the problem to a collision between two point masses. The mass of the striking implement is replaced by its effective mass, the latter quantity depending primarily on the 'swing weight' of the implement and the location of the impact point. For normal or perpendicular impacts, energy loss can be described in terms of the normal coefficient of restitution, e, or an equivalent 'bounce factor', q. For oblique impacts, the outgoing speed and spin of the ball also depends on the tangential coefficient of restitution, e. Examples from baseball, tennis and other sports are highlighted. [ABSTRACT FROM AUTHOR]

Eighteen elite male tennis players were tested to determine their ability to identify string tension differences between rackets strung from 210 N (47 Ib) to 285 N (64 Ib). Each player impacted four tennis balls projected from a ball machine before changing rackets and repeating the test. Eleven participants (6 1%) could not correctly detect a 75 N (17 Ib) difference between rackets. Only two participants (11%) could correctly detect a 25 N (6 Ib) difference. To establish whether varying string tensions affected ball rebound dynamics, the ball's rebound speed and landing position were analysed. The mean rebound ball speed was 117 km h~, with only the trials from the 210 N racket producing significantly lower (P <0.05) rebound speeds than the 235 N and 260 N rackets. This is contrary to previous laboratory-based tests where higher rebound speeds are typically associated with low-string tensions. The anomaly may be attributable to lower swing speeds from participants as they were not familiar with such a low string tension. Ball placement did not appear related to string tension, with the exception of more long errors for the 235 N racket and fewer long errors for the 285 N racket. It was concluded that elite male tennis players display limited ability to detect changes in string tension, impact the ball approximately 6% faster than advanced recreational tennis players during a typical rallying stroke, and that ball placement is predominantly unrelated to string tension for elite performers. [ABSTRACT FROM AUTHOR]

Measurements are presented of the friction force acting on a tennis ball incident obliquely on the strings of a tennis racket. This information, when combined with measurements of ball speed and spin, reveals details of the bounce process that have not previously been observed and also provides the first measurements of the coefficient of sliding friction between a tennis ball and the strings of a tennis racket. At angles of incidence less than about 40° to the string plane, the ball slides across the strings during the whole bounce period. More commonly, the ball is incident at larger angles in which case the ball slides across the string plane for a short distance before gripping the strings. While the bottom of the ball remains at rest on the strings, the remainder of the ball continues to rotate for a short period, after which the ball suddenly releases its grip and the bottom of the ball slides backwards on the string plane. The bounce angle depends mainly on the angle of incidence and the rotation speed of the incident ball. Differences in bounce angle and spin off head-clamped and hand-held rackets are also described. [ABSTRACT FROM AUTHOR]

Tennis courts are normally classified as fast or slow depending on whether the coefficient of sliding friction (COF) between the ball and the surface is respectively small or large. This classification is based on the fact that the change in horizontal ball speed is directly proportional to the COF if the ball is incident at a small angle to the horizontal. At angles of incidence greater than about 16° it is commonly assumed that the ball will roll during the bounce, in which case one can show that the ratio of the horizontal speed after the bounce to that before the bounce will be 0.645 regardless of the angle of incidence or the speed of the court. Measurements are presented showing that (a) at high angles of incidence, tennis balls grip or ‘bite’ the court but they do not roll during the bounce, (b) the bounce:speed ratio can be as low as 0.4 on some courts and (c) the normal reaction force acts through a point ahead of the centre of mass. An interesting consequence is that, if court A is faster than court B at low angles of incidence, then A is not necessarily faster than B at high angles of incidence. An exception is a clay court which remains slow at all angles of incidence. The measurements also show that the coefficient of restitution for a tennis ball can be as high as 0.9 for an oblique bounce on a slow court, meaning that the ball bounces like a superball in the vertical direction and that slow courts are fast in the vertical direction. [ABSTRACT FROM AUTHOR]