Engineering tennis – slowing the game down

Engineering tennis ± slowing the game down

S. J. Haake, S. G. Chadwick, R. J. Dignall, S. Goodwill and P. Rose1

Department of Mechanical Engineering, The University of Shef®eld, Mappin Street, Shef®eld, S1 3JD, UK1 The International Tennis Federation, Bank Lane, Roehampton, London, SW15 5XZ, UK

AbstractAnalysis of results from the four major tennis tournaments shows that the percentage oftie breaks in the men's game has been increasing over the last 30 years. It is hypothesisedthat this is due to the increasing speed of the serve in the game. There was found to be asigni®cant 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 nodifference in the number of tie-breaks between court surfaces.

A larger tennis ball was assessed to see its effect in slowing the game down. Standardand 6% larger pressurised tennis balls were used in experiments to study impacts with a®xed and a freely suspended tennis racket. The coef®cient of restitution of the larger ballwas found to be larger in the ®xed racket tests and analysis of a serve showed that thelarger ball would be served marginally faster than a standard sized ball.

Drag forces on tennis balls in ¯ight were analysed by mounting tennis balls in a windtunnel at wind speeds up to 66.6 ms)1 (150 mph). It was found that different brands ofstandard size tennis ball and a larger tennis ball had a drag coef®cient of approximately0.55. Raising or reducing the nap of the ball changed the drag coef®cient by about10%. Impact experiments of tennis balls on court surfaces showed that the larger andstandard tennis balls rebounded at approximately the same speed at 70% of impactspeed on acrylic and 64% of impact speed on clay. Both sizes of ball bounced steeperoff clay than on acrylic. It appeared that the larger ball rebounded steeper than thestandard ball, although evidence for this was clouded by considerable scatter in thedata.

A computer trajectory program was used to analyse simulated ®rst and second servesat nominally 53.3 ms)1 (120 mph) and 40 ms)1 (90 mph). It was found that a larger ballwould increase travel time to the baseline by approximately 10 ms for a ®rst serve and upto 16 ms for a second serve. This increase was found to be just less than half thatbetween acrylic and clay for the same ball. Travel time is increased further if the ball isincreased in diameter. It was concluded therefore that the introduction of a larger ballcould slow the game of tennis for all strokes and increase the time available for thereceiver to return the ball.

Ó 2000 Blackwell Science Ltd · Sports Engineering (2000) 3, 131±143 131

Correspondence address:Department of Mechanical Engineering,The University of Shef®eld, Mappin Street,Shef®eld, S1 3JD, UK.Tel.: +44 114 222 7739. Fax: +44 114 222 7853.E-mail: s.j.haake@shef®eld.ac.uk

Introduction

The game of tennis, in one form or another, hasexisted for many centuries. Although the game hasseen constant change over the last 125 years, criticsclaim that recent changes to the racket have causedthe game to be dominated by the serve.

Many solutions to slow down the serve have beenproposed, including changing the court surface, theracket, the rules or the ball. Changing the courtsurface is likely to be unacceptable to the tourna-ment organisers; changing the rules is not popularwith the players and limiting the racket even if itcould be done, would not be popular with themanufacturers. This leaves modi®cations of the ballas the most realistic solution (Brody 1996).

This paper will look at some of the evidence toshow that the game is becoming `too fast' and willassess the effect that the introduction of a largerball would have on a serve.

Analysis of the speed of the game

Critics have argued for a number of years that thegame of tennis has become too fast and that matcheshave become dominated by the serve. It has beensuggested that the speed of the game (and thereforedominance of the serve) could be measured bycounting the fraction of tie-breaks (Brody 1990).The dominant servers would win their own games ina set which would lead to a 6±6 score line. Thus, tie-breaks would become more prevalent on fast sur-faces with players whose serve is much better thantheir return game and as players progress further in

tournaments. Figure 1 shows the percentage of setsending in a tie-break against the maximum servespeeds for men at Wimbledon. The serve speeds arethose for 1999 while up to the 10 previous years havebeen used to count the fraction of tie-breaks for eachplayer. It can be seen that there is a general upwardtrend when the serve speed reaches 120±130 mph,supporting the hypothesis that the percentage oftie-breaks is related to serve speed.

Figure 2 shows the percentage of sets that go to atie-break at the four major tennis tournaments ± theFrench Open, Wimbledon, the US Open and theAustralian Open ± for both men and women. It canbe seen that the men's game has a higher percentageof tie-breaks than the women's game at all tourna-ments. The discontinuities in the men's tie-breakdata at the US Open and the Australian Open showwhere grass surfaces were changed to hard courtsurfaces. The fraction of tie-breaks in the men's

Figure 1 Percentage of sets ending in a tie-break vs. serve speedfor the men's game at Wimbledon (tie-break data for last10 years; speed data for 1999 only).

Nomenclature

mr, mb mass of racket and ball, respectivelyvr, vs, vb velocity of racket centre of mass, impact point on strings and ball, respectivelyI moment of inertia of racket about its centre of massx angular velocity of racket about its centre of masse coef®cient of restitution of ball on a clamped racketd distance of impact point from centre of mass of racketCD, CL lift and drag coef®cients for a tennis ballq density of airD diameter of tennis ball

Tennis ± slowing the game · S. J. Haake et al.

132 Sports Engineering (2000) 3, 131±143 · Ó 2000 Blackwell Science Ltd

game is increasing on all surfaces implying thatserves are becoming more dominant. The results forthe women's game are inconclusive, however, withthe number of tie breaks increasing at the US andAustralian Opens, decreasing at Wimbledon whileremaining static at the French Open.

Figure 3 shows the percentage of sets ending ina tie-break for the different surfaces of the majortournaments, averaged over the last seven years(Coe 1999). The error bars indicate standarddeviations. It is generally acknowledged that grassis the `fastest' surface while clay is the slowest withhard-court surfaces somewhere in between (ITF1997). It can be seen that the percentage of tie-breaks mirror this perception within the men'sgame but not within the women's. Outside onestandard deviation, there is a difference betweenthe three lowest percentage tie-break surfaces andthe two highest. The standard deviations for the

women's game indicate that there is no statisticaldifference or trend in the women's game across thedifferent surfaces.

It is possible that the men's game has reached athreshold in the speed of the serve where reactiontimes are longer than the time available to returnthe serve. There is some evidence that a serve of120 mph (53.3 ms)1) is approaching the limit ofhuman reaction time (Braden 1999). The data ontie-breaks for the women's game suggests that thislimit has not yet been reached. This coincides withthe fact that women serve, in general, at less than120 mph.

The only evidence that is freely available is tie-break data since other historical data such as thenumber of aces and speed of serve has not beenarchived or has not been available. In summary,Figs 2 and 3 suggest that the men's game is gettingfaster and that there is a clear difference between

Figure 2 Percentage of sets ending in a tie-break for the men's and women's game at the French Open, Wimbledon, the US Openand the Australian Open.

S. J. Haake et al. · Tennis ± slowing the game


fast surfaces such as grass and slow surfaces such asclay. The next sections of this paper study the wayin which a larger ball can be used to increase thetime available for a receiver to return a serve.

Tennis ball impact with the racket

There has been much work on ball/racket interac-tions over the last 30 years. Much of the work hasconcentrated upon the effect of grip tightness onthe rebounding ball (Hatze 1976; Baker & Putnam1979; Grabiner et al. 1983; Liu 1983; Missavageet al. 1984).

One of the conclusions of this work is that, undersimulated conditions at least, the ball impact withthe racket is much shorter (about 4 ms) than theperiod of oscillation of the racket if it were rigidlyclamped at its handle (about 17 ms) (Brody 1979;Cross 1999). This simpli®es the analysis of tennisball impacts since the racket can be suspendedfreely to simulate a player, while clamping theracket head to a concrete block can be used to lookat the interactions of the ball with only the strings.

Experimental set up for tennis ball impactswith a racket

A Head Prestige Classic 600 tennis racket wasstrung at 65 lb tension (290 N). A compressed air

cannon was used to project standard and largersized pressurised tennis balls normal to the centreof the racket face with the racket either freelysuspended or rigidly clamped about the frame to aconcrete block. Although the string tension was notmeasured after the impact tests, it is likely thatit would have decreased. The impact tests werealternated between balls to take this into account.

The larger tennis balls were manufactured togive the same internal pressure and overall stiffnesscharacteristics as the standard tennis balls. Theaverage size and mass of the balls are shown inTable 1. The larger balls were manufactured toconform to the rules of tennis in mass, compressionand ball rebound from 100 inches (2.54 m).

Results

Figure 4(a) shows that the rebound speed from afreely suspended racket increases with impactspeed. Figure 4(b) shows the apparent coef®cientof restitution (sometimes termed ACOR) which isthe rebound speed of the ball divided by its impactspeed. This does not take into account the recoil ofthe racket itself and so is termed `apparent'. It canbe seen that the apparent coef®cient of restitutiondecreases with impact speed, indicating higher per-centage energy loss at higher speeds. The scatterin the data suggests that there is no differencebetween the large and standard sized balls, althoughone might argue that the larger ball tends to reboundat higher speed.

Figure 5 shows the rebound velocities and coef-®cient of restitution for the standard and largerballs with the ®xed racket. The difference betweenthe large and standard sized tennis balls is moreevident than in Fig. 4. It can be seen, particularly inFig. 5(b), that the larger ball rebounds at a higher

Table 1 Size and mass of standard and large tennis balls usedin impacts with a tennis racket

Diameter (mm) Mass (g)

Standard 64.8 57.6

Large 69.0 59.5

Difference 4.2 (6.5%) 1.8 (3%)

Figure 3 Percentage of sets ending in a tie-break for the men'sand women's game averaged over the last seven years of use forthe different surfaces at the French Open, the US Open, theAustralian Open and Wimbledon.



speed than the standard sized ball from a clampedracket. One possible reason is that the larger tennisball has a thinner shell (2.9 mm compared to3.2 mm) to ensure that the mass is the same as astandard sized ball. Thus, when the ball deforms onimpact, less volume of rubber is deformed causingless work to be done on the ball. Similarly, since theradius of curvature of the ball's shell is larger, it hasa shallower angle through which it is deformed.This would further tend to reduce the work doneon the ball. Since less work is done on the ball, lessenergy is lost through hysteresis in the rubber andthe larger ball rebounds faster than the standardball.

An important point is whether the larger ballwould be launched faster off a racket for a givenserve. The effect of a larger sized ball can beassessed by considering the ball as stationary andthe racket moving towards the ball with thevelocity vector of the racket normal to the racket

face. A rigid body model of the ball/racket impactwas developed to allow velocities from a serve tobe assessed and is described in the followingsection.

A rigid body model of the racket/ball impact

The impact between a ball and a racket can beconsidered as a rigid racket hitting an elastic solidof known coef®cient of restitution to account forhysterisis loss during impact (Brody 1997).

Conservation of momentum gives:

mrvr �mbvb � mrv0r �mbv0b �1�

where mr and mb are the mass of the racket andballs, respectively, vr and vb are the velocity of theracket and ball, respectively, vr

0 and vb0 their

velocity after impact (in a serve vb is set to zero).Conservation of angular momentum gives:

Figure 4 (a) Rebound velocity and (b) apparent coef®cient ofrestitution (ACOR) for standard and 6% larger tennis ballsimpact on a freely suspended tennis racket.

Figure 5 (a) Rebound velocity and (b) coef®cient of restitution(e) for standard and 6% larger tennis balls impact on a tennisracket rigidly ®xed around the frame.



mbvbd � Ix � mbv0bd � Ix0 �2�where I is the moment of inertia of the racket andx the racket's angular velocity about its centre ofmass.

The coef®cient of restitution is given by:

e � �v0b � v0s�=�vb � vs� �3�where vs is the velocity of the impact point on thestring plane of the tennis racket. It is possible tosolve these equations for vb

0, the velocity of the ballafter impact, giving:

v0b �vb Ieÿ I mb

mrÿmbd2

� �� I vr � xd� � 1� e� �

mbd2 � I mbmr� I

� ��4�

The theory was used to calculate the reboundvelocity of a standard sized pressurised tennis ballfrom the same tennis racket as in the previous testsbut freely suspended. The mass of the racket was343 g and the moment of inertia was 0.01534 kg m2

about the centre of mass 320 mm from the buttend. The coef®cient of restitution was calculatedfor all velocities by ®tting second order polynomialregression lines to experimental data similar to thatin Fig. 5.

Figure 6(a), compares the rigid body modelrebound velocity vb

0 for the ball and the racket withexperimental values determined using high speedvideo. It can be seen that the rigid body modelcompares well with the experimental data forvelocities up to approximately 50 m s)1. Themodel can be used therefore with some con®denceup to 50 m s)1 impacts to determine the effect ofracket and ball parameters on the serve of a ball.This is done by giving the racket an initial swingvelocity while keeping the ball stationary.

Figure 6(b) shows the ball serve velocity for aracket swing with no rotational velocity but allowedto freely rotate after impact. The theoreticalanalysis uses the experimentally determined coef®-cient of restitution in Fig. 5. It can be seen that thelarger ball is launched fractionally faster than thestandard sized ball due to its larger coef®cient ofrestitution. The maximum difference of approxi-

mately 0.8 m s)1 (1.8 mph) occurs at a racket planevelocity of 40 m s)1. Although this difference is notvery large it will be taken into account whenconsidering complete trajectories of tennis balls.The next section considers the aerodynamics oftennis balls to allow this to be carried out.

Aerodynamics

Introduction

Ironically, little has been carried out since Newtondiscussed the motion of a tennis ball in ¯ight withOffenburg in 1671 (Thompson 1910) and LordRayleigh wrote his paper `On the irregular ¯ight of atennis ball ' (Rayleigh 1877). Stepanek (1988) carriedout some work on a pressureless tennis ball todetermine lift and drag values at combinations ofspeed and spin. Stepanek found that lift and dragcoef®cients were independent of linear velocity andfound values for CD between 0.55 and 0.75, and

Figure 6 (a) Experimental (circles) and theoretical (lines)rebound velocities from a freely suspended racket for apressurised and pressureless ball; (b) theoretical reboundvelocity from a racket for a standard and 6% larger tennis ball.



between 0.075 and 0.275 for CL, depending uponthe ratio of linear to rotational velocity. The dragand lift force FD and FL for a spinning object arecharacterised by the following formulae (Mehta1985):

FD � 0:5qACDv2 �5�FL � 0:5qACLv2 �6�

where q is the density of the medium travelledthrough, A is the projected area of the object, CD

and CL are the drag and lift coef®cients, and v is thelinear velocity of the object. Previous approacheshave used wind tunnels to determine the drag andlift coef®cients for speeds and spins related to thesport in question. The coef®cients are thereforefunctions of both the linear and rotation velocitiesof the ball. Trajectories can be calculated in aniterative fashion using the following equations:

m�x � ÿFD cos hÿ FL sin h �7�m�y � ÿFD sin h� FL cos hÿmg �8�

tan h � _y

_x�9�

In aerodynamics studies, drag coef®cients arerelated to Reynold's number (Re) by dimensionalanalysis. Re is a dimensionless number relating thespeed of the ¯uid ¯owing over the object and thegeometry of the object. Re is given by:

Re � qvD=l �10�where D is the diameter of the ball and l is theviscosity of the air.

Experiments to calculate CD for tennis balls

A tennis ball was placed in the 5� ´ 4 ft(168 ´ 122 cm) working section of the Markhamwind tunnel at the University of Cambridge, UK.The wind tunnel is capable of wind speeds up to67 ms)1 corresponding to a Reynold's number for atennis ball of approximately 2.8 ´ 105.

Tennis balls were supported by a `sting' whichsupported the balls along the line of the wakebehind the air ¯ow to minimise disruption of theair¯ow. The sting is attached to force transducers

by three wires which allows the calculation of threedimensional forces on the ball. The system was®rst calibrated for all velocities without a ball toallow for forces experienced by the sting alone.Nonspinning tests were carried out to simplify theexperiments and so only drag coef®cients werecalculated.

Results

Drag coef®cients were calculated using a rear-ranged form of eqn (4). Figure 7(a) shows drag

Figure 7 (a) Drag coef®cients for standard and larger pres-surised tennis balls and an unpressurised tennis ball; (b) dragcoef®cient of a standard tennis ball with raised and worn nap.



coef®cients for a standard pressurised tennisball (D � 66.3 mm), a large pressurised tennis ball(D � 67.6 mm), and a pressureless tennis ball(D � 66.0 mm) up to Re � 2.5 ´ 105 (approxi-mately 62 ms)1 or 140 mph). These tennis ballswere slightly different in size to those previouslytested but had the same felt covering.

It can be seen that the CD values for all threetennis balls is approximately a constant of 0.55 forall Reynold's numbers. Although the pressurisedand pressureless tennis balls were different brands,the exterior cloth is very similar and thus it is nosurprise that the drag characteristics are similar.This also holds for the larger tennis ball indicatingthat the air ¯ow across it is similar to the smallertennis balls.

Figure 7(b) shows drag coef®cients for a stan-dard tennis ball with a raised and a worn nap. Thenap was raised manually while wear was simulatedby shaving it using standard hair clippers. It can beseen that these two extremes either raise or lowerthe drag coef®cient by approximately 10% or 0.05.Figure 8 shows streamlines taken at a Reynold'snumber of 6.3 ´ 103 (approximately 1.5 ms)1) for aworn nap and a raised nap. The streamlines for theworn tennis ball (Fig. 8a) ¯ow almost horizontallyindicating that boundary layer separation is justupstream of the poles of the tennis ball. Thetrailing vortices occur within one ball diameter. Itappears that the boundary layer separates earlywhen the nap is raised (Fig. 8b) and that thestreamlines no longer ¯ow horizontally. Thisproduces a larger low pressure region behind theball and consequently a larger drag coef®cient.

In summary, the drag coef®cient can vary by10% if the nap is raised or lowered drastically.Tennis balls of different brands appear to have asimilar, relatively constant, drag coef®cient ofapproximately 0.55.

Tennis ball impacts with the court

Introduction

There has been very little analysis of tennis ballimpacts with court surfaces. Haake (1997; 1998)

carried out some preliminary tests on tennis ballimpacts with an acrylic court surface using highspeed video and a projection machine capable ofprojecting spinning tennis balls at up to 45 ms)1

(101 mph). It was found that a pressureless ballbounced faster and lower than a pressurised tennisball. The effect of ball diameter, with otherwiseidentical characteristics however, has not previouslybeen determined and thus a series of impactexperiments were carried out.

Experimental set up to measure ball impactswith a court surface

Tennis balls were projected using a `Jugs' ballprojector capable of speeds up to 45 ms)1 directly

Figure 8 Smoke streamlines at Re � 6.3 ´ 103 around astandard tennis ball with (a) worn and (b) raised nap.



at a clay and acrylic tennis court surface in theregion of the service line. The impact was recordedusing a Kodak HS4545 video camera running at9000 frames per second.

The `pace' of the surface was also measured usingthe International Tennis Federation sports surfacetest (ITF 1997). In this test, a nonspinning standardpressurised tennis ball is projected at 30 ms)1 at16° to the horizontal. Velocities and angles arerecorded using a light beam arrangement (ITF1997) resulting in a `pace' rating calculated usingthe following equation:

Pace � 100� f1ÿ ��vx ÿ v0x��vy ÿ v0y��g �11�The pace rating categorises surfaces into slow(a rating of 0±35), medium (30±45) and fast (> 40).

Standard pressurised (D � 64.8 mm) and largepressurised (D � 69.9 mm) tennis balls were pro-jected at a clay and acrylic surface at 16° to thehorizontal at 33.7 ms)1 (76 mph) and 44.2 ms)1

(99 mph). Three impacts were carried out at eachsetting for each ball and all apparatus was movedafter each shot to ensure that the surface was notaffected by a previous impact.

Results

Figure 9(a) shows the variation of absoluterebound velocity with increasing impact velocityfor standard and larger tennis balls on clay andacrylic. It can be seen that the ball rebounds fasteron acrylic than on clay and, that given the scatterof the data, there is no signi®cant differencebetween the standard and larger tennis balls. Thedata is converted to the ratio of rebound/impactvelocity in Fig. 9(b) to remove the effect ofvariations from the bowling machine in the impactvelocity. It can be seen that the balls rebound witha ratio of around 0.7 on acrylic and 0.64 on clay.Figure 9(c) shows the ratio of rebound angle toimpact angle vs. impact velocity. A ratio of unityindicates that the rebound angle equals the impactangle. A ratio greater than unity indicates asteeper rebound angle while less than unity ashallower rebound angle than impact. It can beseen that the ball rebounds steeper on clay than

on acrylic. There is no signi®cant differencebetween the standard and larger tennis balls,although the results may suggest that the largerball rebounds a little steeper than the standard

Figure 9 Rebound velocities (a), ratio of rebound velocity toimpact velocity (b), and ratio of rebound angle to impact angle(c) vs. impact velocity for standard pressurised and 6% largerpressurised tennis balls impacting on a clay and an acrylic courtsurface.



sized ball. The scatter in the results indicates thenatural variation in rebound angle for tennis ballsacross a surface.

Discussion

In summary, both the larger and standard tennisballs rebound at approximately the same velocityon each surface. The tennis balls rebound steeperand slower on the clay court than on the acryliccourt. Figure 10(a) shows the ball impact data con-verted to `pace' using eqn (11). The graph showsthat the pace of the acrylic surface is approximatelyconstant (a pace rating of 40±50) with impactvelocity, while the pace decreases (from approxi-mately 40±24) with velocity on the clay surface.This indicates that the clay court is a slowersurface, which may be due to the increasing defor-mation in the court surface as the velocity increases.In eqn (11), this would act to decrease thehorizontal rebound velocity (v0x) and increase thevertical rebound velocity (v0y). This would causethe pace as calculated by eqn (11) to decrease.

The implications of these results for analysis ofa nonspinning impact on a court are summarisedin Fig. 10(b). A ball rebounds with approximately70 � 2% of its impact velocity on acrylic and64 � 1% on clay. The larger ball rebounds at113 � 5% of the impact angle on clay, while thestandard ball rebounds at 105 � 9% of the impactangle. On acrylic, the larger and standard ballsrebound, on average, at 102 � 10% and 97 � 6%of the impact angle. The behaviour of the ball isin¯uenced more by the court surface than thediameter of the ball.

Analysis of a fast serve

The previous sections contain enough informationto allow the analysis of a fast serve using standardpressurised and 6% larger tennis balls onto twosurfaces, clay and acrylic. A program was designedto calculate trajectories of a tennis ball incorpor-ating rebound from a tennis racket, ¯ight throughthe air, impact with a court surface, and ¯ight afterbounce to a receiver standing on the baseline.

Figure 10 (a) Pace rating [eqn (11)] forstandard and 6% larger pressurised tennisballs impacting at 16° on clay and acryliccourt surfaces; (b) average values of ratiosof rebound to impact velocities andangles for clay and acrylic.



The trajectory programme makes the followingassumptions:

1 the rebound of a standard and 6% largerpressurised tennis ball from a racket is foundusing a rigid body model and the coef®cient ofrestitution from a ®xed racket.

2 The ball is nonspinning throughout the traject-ories.

3 CD � 0.55 and the density of the airq � 1.2 kgm)3.

4 The ball rebounds from the surface with thecharacteristics shown in Fig. 10(b).

Figure 11 shows simulated trajectories for astandard and a larger tennis ball with the racketswung in the rigid body model such that the velo-city of the string plane of the racket at the impactpoint was set to either 40.9 ms)1 or 30.2 ms)1.This produced simulated launch velocities of

53.3 ms)1 (120 mph) and 40 ms)1 (90 mph), re-spectively, for the standard ball (i.e. a simulated®rst and second serve). The larger ball is launchedat slightly higher velocities, i.e. 53.7 ms)1

(120.8 mph) and 40.2 ms)1 (90.5 ms)1) due to itshigher coef®cient of restitution off the strings.Figure 11 shows the window of allowable serveslimited by the height of the net or the length of theservice box for the particular serve considered.

It can be seen that the window of possible servesis greater for the slower serve. The angle ofacceptance for good shots for each serve for bothballs is shown in Table 2.

It can be seen that the larger ball allows a slightlylarger acceptance angle as it slows down and dipsmore than the standard ball. The trajectories inFig. 11 are those for an acrylic surface. It can be seenthat for the ®rst serve, the larger ball bounces slightlysteeper and is retarded due to the increased drag.

Figure 11 Trajectories for a®rst (a) and second (b) serve(nominally 53.3 and 40 ms)1) fora standard and a large ball on anacrylic surface showing the win-dow of possible trajectoriesbound by the net and the serviceline. Dashed lines indicate thestandard sized ball; solid linesindicate the larger ball; circlesindicate positions on eachtrajectory at quarter secondintervals.



Figure 12(a) shows the time taken for the balls totravel to the baseline on an acrylic court. Theshaded portion indicates the range of times due tothe angle of the original serve; a shallower anglelands on the service line and arrives at the baselinequicker than the steeper shot just clearing the net.It is seen that a ®rst serve takes approximately0.65 s to reach the baseline, while a second servetakes approximately 0.87 s. The larger ball takes10 ms extra to get to the baseline for a ®rst serveand up to 16 ms for a second serve just clearing thenet. Figure 12(b) shows times to the baseline for astandard ball served onto an acrylic and claysurface. It can be seen that times to the baselineare longer for the clay surface indicating that it isslower. For the ®rst serve the standard ball takes22±26 ms longer on clay than on acrylic. For thesecond serve, clay is slower by 30±40 ms. Thissupports the evidence from the tie-break datapresented in Fig. 3 that clay and acrylic playsigni®cantly differently, although one possibilityis that players intentionally serve at different speedson different surfaces. Comparison of Fig. 12(a) and(b) shows that the larger ball can produce partof the extra time of 22±26 ms that clay givescompared to acrylic for a ®rst serve. It can beconcluded, therefore that the two different ballsshould play signi®cantly differently on the samesurface.

Braden (1999) suggests that players do notperceive the ball until approximately 0.25 s afterthe serve. Thus, for a 53.3-ms)1 (120-mph) serve aplayer has about 0.4 s to react. The evidence of thispaper indicates that the larger ball would giveabout 2�% extra reaction time for a ®rst serve. Thetime available to react could be lengthened further

by increasing the size of the ball to, say, 72 mm(2.825 in) and by reducing its stiffness (or internalpressure) so that it rebounds from the racket at thesame speed as the standard ball. It is possible that alarger ball launched at the same speed as a standardball would take up to an extra 40 ms to reach thereceiver.

The International Tennis Federation has recentlybrought in a rule to allow larger tennis balls to beused in tournaments from 2000 onwards. Player

Figure 12 Travel times to the baseline for a ®rst and secondserve (nominally 53.3 and 40 ms)1) for (a) a standard and largeball on an acrylic surface, and (b) for a standard ball on anacrylic and a clay surface. The shaded portions indicate therange of times corresponding to the range of possible angles ofthe serve.

Table 2 Acceptance angle for simulated ®rst and second servesde®ned by serves just clearing the net and just hitting theservice line (standard and larger ball served at 120 and120.5 mph, respectively)

Standard ball 6% larger ball

First serve 0.93° 0.96°Second serve 1.67° 1.72°



testing will show whether the players' perceptionsmirror the ®ndings of the work described here.

Conclusions

Impacts of standard and 6% larger pressurisedtennis balls with a tennis racket showed that thelarger ball would rebound at a higher speed from aracket during a serve. It was thought that this wasdue to the lower work done on the larger ball andtherefore the lower energy losses during impact. Anaerodynamic study found the drag coef®cients oftennis balls to be approximately 0.55. This valuecan be changed by approximately 10% by raising orlowering the nap. Impacts on a clay and an acrylictennis court showed that the standard and largertennis balls rebounded at approximately the samevelocity; it is suggested that the larger ball re-bounded slightly steeper. The results of thisanalysis were incorporated into a trajectory modelwhich showed that a 6% larger tennis ball wouldgive the receiver approximately 10 ms more reac-tion time when facing a ®rst serve, and up to 16 msmore for a second serve. This time difference wasfound to be just less than half that between thesame ball on an acrylic and a clay surface. Thisreaction time could be increased if the larger ball ismanufactured to rebound from a racket at the samespeed as a standard ball. It is concluded, therefore,that the introduction of a 6% larger tennis ballcould increase the time available to a player tomake a service return, thereby engineering thegame to slow it down.

Acknowledgements

The authors would like to thank the InternationalTennis Federation (ITF) for sponsoring theprojects detailed in this paper. Thanks also goesto the Alison Cooke and the University ofCambridge for help with the aerodynamicprojects, and Howard Brody and Rod Cross formany useful discussions during the preparation ofthis paper.

References

Baker, J. & Putnam, C. (1979) Tennis racket and ballresponses during impact under clamped and free-stan-ding conditions. Research Quarterly, 50, 164±170.

Braden, V. (1999) Vic Braden Newsletter for 6/26/99,www.vicbraden.com.

Brody, H. (1979) Physics of a tennis racket. AmericanJournal of Physics, 47, 482±487.

Brody, H. (1990) Predicting the likelihood of tie-breaks.TennisPro, March-April, 21.

Brody, H. (1996) A modest proposal to improve tennis.TennisPro, January-February, 6±8.

Brody, H. (1997) The physics of tennis III: the ball±racketinteraction. American Journal of Physics, 65, 981±987.

Coe, A. (1999) The balance between technology andtradition in tennis. International Tennis Federation Tech-nical Commission, Report. International Tennis Federa-tion, Roehampton, London, UK, p. 42.

Cross, R. (1999) The sweet spots of a tennis racket. SportsEngineering, 1, 63±78.

Grabiner, M., Groppel, J.L., and Campbell, K.R. (1983)Resultant tennis ball velocity as a function of off-centerimpact and grip ®rmness. Medicine and Science in Sport andExercise, 15, 542±544.

Haake, S.J. (1998) Impact problems of balls in Tennis andGolf. Transactions of the JSME, 64±623, 2318±2327.

Haake, S.J. & Goodwill, S.R. (1997) Tennis ball impacts onsynthetic turf, JSME Symposium on Sports Engineering,97±34, 137±143.

Hatze, H. (1976) Forces and duration of impact and griptightness during the tennis stroke. Medicine and Science inSports, 8, 88±95.

ITF (1997) An initial ITF study on performance standardsfor tennis court surfaces. In , pp. 41. International TennisFederation, Roehampton, London, UK.

Liu, Y.K. (1983) Mechanical analysis of racket and ballduring impact. Medicine and Science in Sport and Exercise,15, 388±392.

Mehta, R. (1985) Aerodynamics of sports balls. AnnualReview of Fluid Mechanics, 17, 151±189.

Missavage, R., Baker, J. & Putnam, C. (1984) Theoreticalmodelling of grip ®rmness during ball racket impact.Research Quarterly for Exercise and Sport, 55, 254±260.

Rayleigh, Lord (1877) On the irregular ¯ight of a tennisball. Messenger of Mathematics, 7, 14±16.

Stepanek, A. (1988) The aerodynamics of tennis balls ± thetopspin lob. American Journal of Physics, 56, 138±141.

Thompson, J.J. (1910) The dynamics of a golf ball. Nature,85, 2151±2157.



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Engineering tennis – slowing the game down