Analysis of results from the four major tennis tournaments shows that the percentage of tie breaks in the men’s game has been increasing over the last 30 years. It is hypothesised that this is due to the increasing speed of the serve in the game. There was found to be a significant difference in tie breaks between slower clay surfaces and faster grass surfaces. The women’s game, on the other hand, showed no increase in tie‐breaks and no difference in the number of tie‐breaks between court surfaces. A larger tennis ball was assessed to see its effect in slowing the game down. Standard and 6% larger pressurised tennis balls were used in experiments to study impacts with a fixed and a freely suspended tennis racket. The coefficient of restitution of the larger ball was found to be larger in the fixed racket tests and analysis of a serve showed that the larger ball would be served marginally faster than a standard sized ball. Drag forces on tennis balls in flight were analysed by mounting tennis balls in a wind tunnel at wind speeds up to 66.6 ms−1 (150 mph). It was found that different brands of standard size tennis ball and a larger tennis ball had a drag coefficient of approximately 0.55. Raising or reducing the nap of the ball changed the drag coefficient by about 10%. Impact experiments of tennis balls on court surfaces showed that the larger and standard tennis balls rebounded at approximately the same speed at 70% of impact speed on acrylic and 64% of impact speed on clay. Both sizes of ball bounced steeper off clay than on acrylic. It appeared that the larger ball rebounded steeper than the standard ball, although evidence for this was clouded by considerable scatter in the data. A computer trajectory program was used to analyse simulated first and second serves at nominally 53.3 ms−1 (120 mph) and 40 ms−1 (90 mph). It was found that a larger ball would increase travel time to the baseline by approximately 10 ms for a first serve and up to 16 ms for a second serve. This increase was found to be just less than half that between acrylic and clay for the same ball. Travel time is increased further if the ball is increased in diameter. It was concluded therefore that the introduction of a larger ball could slow the game of tennis for all strokes and increase the time available for the receiver to return the ball.

The dynamic properties of six types of tennis balls were measured using a force platform and high-speed digital video images of ball impacts on rigidly clamped tennis rackets. It was found that the coefficient of restitution reduced with velocity for impacts on a rigid surface or with a rigidly clamped tennis racket. Pressurized balls had the highest coefficient of restitution, which decreased by 20% when punctured. Pressureless balls had a coefficient of restitution approaching that of a punctured ball at high speeds. The dynamic stiffness of the ball or the ball-racket system increased with velocity and pressurized balls had the highest stiffness, which decreased by 35% when punctured. The characteristics of pressureless balls were shown to be similar to those of punctured balls at high velocity and it was found that lowering the string tension produced a smaller range of stiffness or coefficient of restitution. It was hypothesized that players might consider high ball stiffness to imply a high coefficient of restitution. Plots of coefficient of restitution versus stiffness confirmed the relationship and it was found that, generally, pressurized balls had a higher coefficient of restitution and stiffness than pressureless balls. The players might perceive these parameters through a combination of sound, vibration and perception of ball speed off the racket.

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* Corresponding author ABSTRACTThe aerodynamic properties of an association football were measured using a wind tunnel arrangement. A third scale model of a generic football (with seams) was used as well as a 'mini-football'. As the wind speed was increased, the drag coefficient decreased from 0.5 to 0.2, suggesting a transition from laminar to turbulent behaviour in the boundary layer. For spinning footballs, the Magnus effect was observed and it was found that reverse Magnus effects were possible at low Reynolds numbers. Measurements on spinning smooth spheres found that laminar behaviour led to a high drag coefficient for a large range of Reynolds numbers and Magnus effects were inconsistent, but generally showed reverse Magnus behaviour at high Reynolds number and spin parameter. Trajectory simulations of free kicks demonstrated that a football that is struck in the centre will follow a near straight trajectory, dipping slightly before reaching the goal, whereas a football that is struck off centre will bend before reaching the goal, but will have a significantly longer flight time. The curving kick simulation was repeated for a smooth ball, which resulted in a longer flight time, due to increased drag, and the ball curving in the opposite direction, due to reverse Magnus effects. The presence of seams was found to encourage turbulent behaviour, resulting in reduced drag and more predictable Magnus behaviour for a conventional football, compared to a smooth ball.

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Modern tennis rackets are manufactured from composite materials with high stiffness-to-weight ratios. In this paper, a finite element (FE) model was constructed to simulate an impact of a tennis ball on a freely suspended racket. The FE model was in good agreement with experimental data collected in a laboratory. The model showed racket stiffness to have no influence on the rebound characteristics of the ball, when simulating oblique spinning impacts at the geometric stringbed centre. The rebound velocity and topspin of the ball increased with the resultant impact velocity. It is likely that the maximum speed at which a player can swing a racket will increase as the moment of inertia (swingweight) decreases. Therefore, a player has the capacity to hit the ball faster, and with more topspin, when using a racket with a low swingweight.

There has been little three-dimensional (3D) analysis of the interaction of a tennis ball and racket during realistic play conditions. This paper is a descriptive study of elite players in practice conditions. The method used records racket and ball movement in 3D, intrudes minimally into the player's environment and has a high level of portability. Testing was performed using two Phantom V4.2 high speed video cameras operating at 1,000 frames per second. Racket movement was tracked using five reflective markers attached to the player's racket and the ball was tracked as a single point. The method allowed accurate measurement of ball and racket speeds, impact positions, and angular velocities of the racket in threedimensions. It was used at the 2006 Wimbledon qualifying tournament in practice conditions to record 106 shots from 16 internationally ranked players. The results obtained showed that all players aim to hit the node point on the racket face in a standard forehand drive. The average postimpact ball velocity of male players was 9.4% greater than that of female players at 33.9 m s -1 , post-impact ball spin was 22.3% higher at 1,480 rpm. These results could be used to confirm previous research into player movement and impact, or as a basis for future investigation into the interaction between the ball, racket and player.

A model has been derived that determines the ball, stringbed and racket frame motion for an impact between a tennis ball and racket. This paper describes the model and methods used to verify it experimentally. The model incorporated parameters such as racket mass, moment of inertia and ball stiffness. The work was conducted to produce a tool that could be used to identify the importance of each of the parameters on the impact. The ball was modelled as simple spring and damper in parallel while the stringbed was modelled as a spring in series with the ball. The values of these spring and damper parameters were determined experimentally. It was assumed that the racket frame was a rigid body to simplify the analysis. Two different methods of supporting the racket were modelled, and the balls were always projected perpendicular to the string plane. Firstly, the racket was head-clamped and all the balls impacted at the geometric string centre of the racket. Good agreement was found between the experiment and model data for both ball rebound velocity and maximum stringbed deflection during impact. The second method of support involved freely suspending the racket at the tip on a small pin. Three different impact positions along the longitudinal axis of the racket were tested. Good agreement between the experiment and model data was found for the ball rebound velocity for impacts at the geometric string centre of the racket. The model over-predicted the experimental racket rebound velocity (generally by less than 5 per cent) for all impact positions. A qualitative analysis assigned this small difference to the assumption that the frame was a rigid body and therefore vibration losses were not accounted for.