Racquets cial rule ⅲ tennis about racquet weight. You could use a 1 OOO-gram racquet if you wanted to, as heavy as a baseball bat, but baseball players don't have to run around with their bat chasing after the ball. lt doesn't hurt tO try a heavier racquet tO see if it makes any difference to your game. The simplest way t0 d0 that is t0 add some lead tape t0 the tip and/or the handle or to borrow another racquet. You might find, for example, that you struggle t0 swing the racquet comfortably, or you might find that you can get a bit more power ⅲ your shots. Ten grams added to the tip of a racquet will feel quite different [ 0 an extra 10 grams ⅲ the handle. The effect is described in the section on swingweight. LIGHT S. HEAVY RACQUETS If a heavy and a light racquet are each swung atthe same speed, the ball will come Off the heavy racquet faster because the heavy racquet has more momen- tum and 1 れ ore energy that it can transfer tO the ball, and it will lose less ener- gy. However, heavy racquets might not be swung as fast as light racquets. There is, therefore, not a big difference in maximum power between heavy and light racquets. ln general, racquets tend い be swung at medium tO fast pace rather than maximum possible speed because players need [ 0 make sure the ball goes in. ln that case heavy racquets Offer a bit more power and control than light racquets because they don't need tO be swung as fast tO achieve the same ball speed. Outgoing ball speed is a combination 0f rebound speed and racquet speed. The rebound speed is really a measurement Of hOW much energy iS lOSt in the rac- quet and ball collision. The higher the rebound speed, the less energy is lost. The energy available tO use and lose comes from the mass and the motion of the ball and racquet. During the collision, both the ball and the racquetlose energy. The heavier the racquet, the more energy the racquet has available at a given racquet speed and the less energy it loses during the collision. Top players are generally stronger and fitter and can make better use of a heav- ier racquet by swinging it faster than the average recreational player. Conversely, if a player needs to getthe racquetto the ball quickly, a light rac- quet will help. For that reason, even professionals use racquets that are much lighter than they are capable of swinging. ln theory, a heavier racquet should help to reduce arm injuries. There is anec- dOtal evidence from veteran coaches that arm and shoulder injuries increased 33

Racquets Match Point Box 1.3 Maximum TheoreticaI BaII Speed 『 the racquet was petfect and lost no energy, and ifthe ball was a 0 perfect and lost no energy,then the maximum possible serve speed tO racquet speed ratiO would be 2. O. 旧 other words, the serve speed would be twice the racquet speed. に is impossible tO dO any better than that according tO the laws ofphysics, no matter what is done tO improve the racquet frame or the strings. However, the factor 0f2.0 here so requires the racquet to be infinitely heavy. Such a racquet would sink tO the center of the earth as soon as it was constructed, and it would suck in the sun and the moon while it was at it. For racquets around 300 or 400 gm, and for perfect racquets and balls, the maximum theoretical serve speed tO racquet speed ratiO is around に 5. 旧 practice, the serve speed to racquet speed ratio is typically about に 4. M0dern racquets are therefore almost as powerful as they can be, given that the rules specify that balls must lose energy in every collision. That being said, the easiest way to hit the ball faster is still simply to swing faster. Every player has more ⅲ reserve than any amount of fiddling with the rebound power can deliver. But for any given racquet, the rebound power is the way tO compare racquet power. POWER AND ENERGY LOSS 、、 More powerful" actually means less energy loss. So, although racquet adver- tisements are constantly singing the praises Of "more powerful" racquets, these racquets have no propulsion system. AII the energy that is possible is present before the impact. That is the energy of motion in the racquet and ball approaching each 0ther. The impact does not produce energy; it only loses it. Designing a powerful racquet is all about limiting energy loss, not about pro- ducing energy. And that is what is measured ⅲ rebound power. The primary reason the ball rebounds faster on a racquet with a high rebound power is that the extra weight limits these extraneous 1 れ 0 ⅱ ons and thus creates a 1 れ ore sta- ble platform from which the rebound can take place. If you lay a racquet on the court and put your foot on the throat area so that racquet can't recoil or vibrate, then the rebound power will increase tO about 0.9. That is, the ball will bounce い about 80 percent of the drop height (rebound power is the square root of the bounce height ratio). This provides a rather dramatic demonstration Of the fact that rebound power is strongly 25

Strings TENSION AND POWER Experiments with strings in the lab have shown that when a non-deformable hammer strikes any string at almost afiy tension, each string takes in the same amount 0f energy and gives about 95 percent 0f this back. Therefore the ham- mer is ejected at the same speed for every string. The differences are that stiffer (whether by material, tension, or bOth) strings will stretch less and exert a higher force on the hammer, but d0 so for a shorter period 0f time (shorter dwell time). The higher force for less time ejects the hammer atthe same speed as dO the SOfter strings exerting a lesser force for a longer time. Nothing changes ⅲ the string's dynamics when you hitthe strings with a ball instead of a hammer. The strings still give back almost all 0f the energy that goes intO them, but the amount Of energy the string actually receives will depend on how much of the available energy goes into compressing the ball instead. Just as we know from string tests hOW much energy the strings give back, we know from similar ball tests that a ball gives back only 55 percent of the energy that goes int0 it. Energy is divided depending on the relative stiff- ness Of tWO colliding objects—more energy going intO the softer Object. For the hammer and string, the hammer is SO much stiffer than the string that all the energy Of the collision goes intO stretching the strings (the hammer doesn't deform). But the ball and the stringbed are about the same stiffness as each other (the ball compresses the same amount that the stringbed deflects). SO they each will get about half the energy, whereupon the ball loses 45 percent of its share, and the strings lose their five percent. If you change the stringbed stiffness by raising tension or changing tO a stiffer material, then the stringbed will be stiffer than the ball and will not deflect as much, and the ball will com- press more. More energy is directed into the ball, 0f which it obligingly loses 45 percent, and the ejection speed of the ball is less. The opposite happens if you lessen the stringbed stiffness by lowering tension. The strings will take a greater share 0f the impact energy, leaving less for the ball t0 lose. HOW much extra ball velocity are we talking about? lt's not as great as you would think, given the folklore on the subject and player's anecdotal reports. The 01d adage "string 100SC for power, tight for control" is true, just not to the extentthat most people think (). e. , looser strings will not change power by 20 percent, IO percent, or even 5 percent). If you drop string tension by 10 pounds, the percentage gain ⅲ ball velocity will be less than two percent, or about 1.2 mph on a 60 mph groundstroke. / 5

Chapter One Rebound power is the power" that is available due simply t0 the racquet's existence and that iS always available on every swing, ShOt, and situation that the game Of tennis can throw at you. REBOUND POWER THE SUM OF ALL RACQUET PROPERTIES The magic Of rebound power is that lt is the result Of the combined influence of all the frame's physical parameters. The effects of the racquet's mass, bal- ance, swmgweight, flex, headsize, pattern, string, and tension are each accounted for in measuring rebound power. The rebound iS the consequence of the effects of all of these things, as well as every other feature designed into the construction and stringing Of the racquet. A racquet with a higher rebound has a more powerful combination Of these factors at that impact location than a racquet with a lower rebound. If you had a map 0f the stringbed showing the rebound power value at one-inch locations radiating out frOI れ the center Of the racquet, you could compare the power Of every racquet in the area Of your impact zone (indicated by where you find ball fuzz on the strings 0f your rac- quet) ・ Such a rebound power map shows the results of the different flow of energy ⅲ a racquet for each impact location. When the ball strikes a stationary, free- standing or hand-held racquet, the energy 0f the ball is divided between rac- quet translation (linear motion) , rotation, twisting, and bending. Each Of these motions siphons 0ff energy, making it unavailable for propelling the ball. The energy that is left over (elastic energy stored in the stretched string and com- pressed ball) is used t0 rebound the ball off the strings, though about 25 per- cent 0f that energy is lost ⅲ the process also. ln general, lighter, more flexible racquets and stiffer stringbeds result in more energy being wasted and less tO propel the ball. Also, as you move out from the center of the stringbed, rebound power declines, more SO for smaller head racquets than larger ones. Rebound power is actually tricky [ 0 measure accurately without special equip- ment, but the concept is incredibly simple, and quickly lays t0 rest any "which racquet is more powerful" debates. lt is astonishing that the intrinsic, Off-the- shelf power Of the racquet at any given stringbed location is revealed SO com- pletely and accurately by such an unimpressive, simple, slow collision as the drop test shown in Figure 1.3. NO on-court, 。、 realistic" situations are necessary tO determine a racquet's built-in power—the power that is available in every single hit of the ball, independent 0f swing speed. 24

Racquets BaII: 57 gm 57 x 6 = 342 342 x 6 = 2052 Racquet: 342 gm 35 running (Figure 1.13 ). pendulum. A golf swing also has this action, as does throwing and walking or transfer its energy tO the racquet. The action is similar tO that of the double 1 れ um energy intO the racquet, the forearm needs [ 0 SIOW down SO that it can imum speed t00 early, while the forearm is still swinging rapidly. To get maxi- behind and hit the ball い 0 la に If the racquet is too light, it will reach its max- times heavier than the racquet. If the racquet is too heavy, it will tend to lag speed. The 1 れ OSt efficient swing style iS one where the forearm is five or six ferent action because the tip and the handle usually travel at about the same swung at maximum speed and faster than the handle. A volley requires a dif- and the ball. ThiS sequence Of events ensures that the tip Of the racquet is energy flows from your upper arm い your forearm and finally to the racquet to the racquet and を om the racquet t0 the ba / /. 「砒 / 0 0 「 0b0 砒 6 : / gives the best 卩 0 Ⅳ ofenergy om the arm Figure れ ldeal racquet weight. The 0 「 m to racquet mass Arm: 2052 gm

Ba and Bounce Ground Reaction Force on BaIl lncident Path Forward Force on Court Friction Force on Ball Normal Force on Court Figure 3.4 The force ofthe ba ″ on the courtis equal and opposite the force ofthe court on the ba 〃 . The て e on the ba ″ has two component parts: one pushing upward, known as the ground 「 e ロ ( on に e , which causes the b ロ〃こ 0 bounce, and the Other 居 the をに t / on に e , which resists the 「Ⅳ ard motion Ofthe ba 〃 across the court and thus slows down. hardness of the court and the ball. If the court is soft, energy will be lost deforming the court surface ()r worse, if the ball lands ⅲ a pile of dirt, it won't bounce at all). The surface does not spring back fast enough or efficiently enough to aid the ball ⅲ its bounce, so that energy is 、、 lost , ' ' and the ball won't bounce as high. Every surface tends tO lose a characteristic amount Of energy, which determines the vertical bounce speed (and thus height) for that court. Some hard courts, such as those used at the US and Australian Open, are con- structed with a layer of rubber under the top acrylic surface. The acrylic green paint is mixed with sand tO control the surface friction on the ball while the rubber helps cushion the surface under foot. The vertical bounce height off these courts remains relatively high, despite the SOft rubber cushion, because the ball doesn't compress the surface as much as it compresses grass. Hard courts are described as hard because they are much harder than the ball. The ratio of the ball's vertical speed after the bounce to that before is known as the 。、 coefficient 0f restitution" (COR). If the vertical speed after the bounce is faster on one court than another, the ball will bounce higher on that court. COR is about 0.75 for grass, 0.8 for hard courts, and 0.85 for clay courts. The effect of these differences is shown ⅲ Figure 3.5. The ball's vertical bounce will be highest and fastest on clay, lowest and slowest on grass. COR is not constant, however. The efficiency of the bounce deteriorates at higher impact speeds, and COR will be less. Higher speeds cause more ball deformation, which causes energy loss. But, at the same time, as the ball deforms more, it gets stiffer and is harder tO deform. These two effects work in opposite directions such that the net result is that a faster ball will always 9 /

Chapter One affected by recoil and vibration 0f the racquet frame, each of which take ener- gy away from the ball. lt also shows thatthe strings play a very important role in reducing energy loss ⅲ the ball. If the ball is dropped on the court instead of the strings, then the ball will bounce t0 only about 55 percent of the drop height. The strings don't supply any extra energy to the ball, even though it might appear that they d0. Rather, the strings act t0 soften the impact. As a result, the force on the ball is reduced, so the ball compresses by a smaller amount and loses a smaller amount Of energy. At the same time, 95 percent Of the energy required t0 stretch the strings is given back t0 the ball when the strings spring back tO their onginal position. THE BAD NEWS: REBOUND POWER VALUES AREN'T READILY AVAILABLE The problem is that rebound power is not a published parameter listed ⅲ man- ufacturers' marketing materials or on the point-of-sale "face card" that is attached tO the racquet head in the store. This is unfortunate because a stan- dardized rebound power map at selected intervals from the center Of the stringbed would give players an accurate, quantitative way tO compare rac- quets. lt would shOW the most powerful locations on the racquet and hOW rap- idly the power deteriorates as you hit outside of the middle of the string face. lnstead, players must rely on hyperbolized marketing—、 ultimate power"—or, all t00 often, the uninformed, opinionated recommendations Of the store's school-vacation sales staff. The other, better, alternative is tO make judgments Of what rebound power is likely tO be compared tO another racquet based on the racquet features that comprise them—•、 veight, stiffness, balance, headsize, etc. ln general, more weight, more weightlocated farther toward the ends and sides 0f the racquet (higher swingweight, higher balance, and larger/wider headsize) , and stiffer flex all contribute t0 greater rebound power. Also, anything that makes the stringbed softer will increase rebound power, and anything that makes it stiffer will decrease it. Given these generalizations, you can dO a rebound test in your mind, visualizing the effect 0f each racquet property on the rebound. At least that gives you a conceptual framework upon which tO make an intelligent decision. (This conceptual framework will be built up much more in the rest of this book. ) 26

Chapter Three BOUNCE FACTOR #l: ANGLE OF INCIDENCE The one thing we all remember from sch001 is that when an object bounces Off another stationary and immovable object, the angle Of incidence equals the angle Of reflection (see Figure 3.2 ). That statement is only true in tWO ideal sit- uations: ( 1 ) when there is no energy IOSS and no friction, or ( 2 ) when the ver- tical and horizontal speeds 0f the ball decrease by the same fraction. For exam- ple, if the horizontal and vertical speeds both decrease by 20 percent, or if they both decrease by 30 percent, then the angle of reflection will be exactly equal [ 0 the angle Of incidence. However, in 1 れ OSt cases Where a tennis ball bounces off the court, the vertical speed decreases by about 25 percent and the horizon- tal speed decreases by about 30 t0 40 percent. After the bounce, the ball climbs vertically at a relatively fast rate compared with its reduced horizontal speed, SO it bounces at a steeper angle than the incident angle. The angle Of incidence is equal tO the angle Of reflection when light reflects Off a mirror, but not for tennis balls bouncing Off a tennis court. The bounce rule for tennis iS that the angle 0f reflection is almost always greater than the angle 0f incidence. The only exception is when the incident ball lands with a 10t 0f topspin, which we consider later. Angle 0f incidence 20 。 Angle 0f reflection Figure 3.2 lfthe angle ofincidence is 2 〇 degrees, the angle 0 「 reflec- tion is usually greater than 2 〇 degrees. The rebound angle depends on the angle of the incident ball, and it also depends on what happens t0 the ball in the horizontal and vertical directions during the bounce. If the ball comes in at a small angle, it will also bounce off the court at a small angle. If the incident ball approaches the court at a steep angle, the ball will bounce up at a steep angle. When a ball is hit with topspin, れ tends tO dive down ontO the court at a steeper angle. Hence it will tend tO bounce up at a steep angle. Each of these things happen because, when a ball hits the court, a vertical force pushes the ball up and a horizontal force acts tO slow the ball and change its spin. 94

Chapter One Drop Height 20 in. Rebound Height Drop Height Rebound Power 3.2 20 0.4 (middle of strings) 3.2 in. 0.2 20 0 」 ()t tip) 0.2 in. Rebou nd H eight Hand-held racquet Figure 1.3 The ratio 0 「 e rebound height to the drop heightis the energy 「 e 「 n. So, 3.2 / 20 = . / 6 , which means 砒 / 6 percent 0 「 e energy is returned on the rebound ⅲ the middle ofthe strings above. The square 「 00t ofthis num- ber, 0.4 , is the ratio ofthe exit speed to the incoming speed ofthe ba 〃 . The sci- entific name 「 this 田 t / 0 is "Apparent Coefficient ofRestitution. "We ⅣⅢ ca 〃 "Rebound P 〇Ⅳ e 「 . ” power" 0f the racquet (。、 rebound power" being preferable to the scientific term which is 、、 Apparent Coefficient of Restitution," or "ACOR"). Rebound power is the amount of power "built into" the racquet by the manu- facturer. The ultimate amount Of power you can generate will depend on your ability to swing it, but rebound power is what players 100k for (probably with- out realizing (t) when they go to a shop ⅲ search of a powerful racquet. Rebound power varies for each location on the racquet. The location with the highest rebound is also the spot with the fastest rebound. This spot can be called the 、、 maximum rebound point" ()s we will see below, this is not neces- sarily the spot of maximum exit ball speed, however). lt is usually located in the throat region of the racquet. Rebound power depends mainly on the weight of the racquet and how it is distributed around the frame and ⅲ the handle. Rebound power increases with racquet weight, and it is larger when the rac-

Chapter Two energy when struck by a ball. Likewise, when a string loses tension and, according [ 0 some players, goes 、 dead," it doesn't lose power or energy return, as the term would seem tO imply.. The string power is the same ()f not more). We've seen that as tension declines, the string actually takes in, and thus gives back, more energy (which translates to ball velocity). So tension loss does not equal dead" ⅲ terms of "power" (ball velocity) , but ⅲ terms of diminished force of impact, shock, and feedbacl ←—in other words, ⅲ terms of 、、 feel. Consistency of feel is very important. That is why you shouldn't leave your rac- quet in the car on a hOt day—because heat accelerates tension loss, and, no, you can't get it back by sticking it in the refrigerator. TENSION LOSS AND FEEL The effect of tension loss on what you feel depends on several factors: the stiff- ness [ 0 begin with, the stiffness that feels best tO you, your sensitivity tO change, and hOW you interpret what you feel at a given stringbed stiffness com- pared tO another. You don't get the same punch" when you hitthe ball with diminished tension. The ball may go faster and farther, but it feels like less oomph. And because you have actually lost"control" as witnessed by the ball going farther (). e. , not where you are aiming) , you may back off your stroke speed, which lessens the feel of impact oomph even more. So you haven'tlost power, but you have 10S [ the feel of providing the power, being in control, and getting feedback confir- mation from the racquet punch. ln essence, you have lost ShOCk, not power. The impact doesn't shOCk you as much. NOt as much shock is not as much 、 'feedback. '' SO, in one sense, when you choose a string, you are choosing the level of shock that feels good or proper [ 0 you. (However, "feeling good" does- n't mean thatthe level of shock that you like is good for you. ) NO matter hOW you interpret the change in feel as tension goes down, the fact is thatthe feel does change. Strings do stabilize with time, and the rate of loss continually slows down. Consequently, a relatively 。、 fast changing string may still be slow enough to stay within your 。、 feel range" for an acceptable amount of playing time. That will depend on your sensitivity and response to change. Nonetheless, a rule of thumb does apply. Any strings at the same stiffness value should feel close tO the same in the same racquet for a certain amount of time, no matter what amount Of tension IOSS was required tO arrlve at that stiffness. The amount Of tension IOSS is irrelevant as long at the resulting stiffness feels good tO you. SO, tension loss is not in itself a necessarily good or bad thing. 78