The Rocket Science Of Tennis And Its

Racket Essay, Research Paper

Its Not Rocket Science, Its Racket Science

Remember warm Sunday afternoons when everyone loaded into the car with the

intention of spending some family time together? For me, being one of eight children, it

was all too familiar. The entire trip was nothing but fighting over what we were going to

get, and who got to pick the cereal. Once we got to the popular cereal isle my brothers,

sisters, and I were in constant battle deciding between Cocoa Puffs and Trix. What did

we end up with though? Kaboom or something generic like that. It was the same thing

(that is what my parents always said), but it never really tasted the same. Imagine my

fright when I announced that I was going to join the tennis team and I needed a racket.

Just like the cereal, I knew that my parents were interested in saving money. Quality was

not in the budget. I had envisioned the Radical Tour 260, the latest and most practical

tool for the game of tennis. My parents, on the other hand, had different intentions. We

were off to WAL-MART to find the most economical racket that reasonably fit into the

budget. At least, that is how my parents explained the situation. Otherwise, in normal

language, we were going to pick out the cheapest racket on the shelf. I began arguing,

stressing the importance of how the racket affected my skill on the court. I continued

rambling and whining, and with that my father issued a challenge: If I could find

scientific research backing up my reasoning for needing the Radical Tour 260, he would

be sold. My need for that racket was overwhelming. I did not want to be the only guy on

the team without the racket. It just wouldn?t be fair. With that thought, I ran off to the

library to start researching. This, my report, is what I gave my parents the next evening.

To determine how important the racket is in the success of a tennis player, one must first

understand the basic motions of the ball, the many swings affecting the ball, the anatomy

of the racket, and how, through the laws of physics, the racket and its actions can be

manipulated to ensure success in even the beginning tennis player. To achieve a full

understanding of how physics affects the game of tennis, I will begin with defining a few

basic physical principles that influence motions of the ball. Next, I will apply these

definitions to several physical characteristics such as the coefficient of friction, speed,

resistance, Newton?s Laws, Magnus force, gravitational pull, and the conservation of

momentum. Finally, I will use these characteristics to describe how and why the

technology of tennis rackets has changed in recent years.

The motion of a tennis ball through air is determined by the laws of physics. The

way in which the ball goes over the net on a serve is not as simplistic as it might sound.

It includes velocity (both final and initial), acceleration of the ball, forces acting on the

ball and the angles of motion during the swing and the follow through. Speed is a ratio

between the displacement divided by the time it took for the displacement to occur

(v=d/t). For example, imagine a tennis player hits a ball ten yards in two seconds. The

average speed of the ball is five yards per second. At some point, the ball may have been

going faster or slower than five yards per second, but again it is the average speed. When

the velocity of the ball changes, the ball undergoes acceleration. Acceleration is the

change in velocity divided by the interval of time. When the tennis ball?s velocity and

acceleration are in the same direction, the speed of the ball occurs with time. When the

ball?s velocity and acceleration are in opposite directions, however, the speed of the ball

decreases with time.

Once the ball is first shot into the air, the laws of physics take over and determine

where it will go. There is nothing that the player or his or her opponent can do to guide it

or change its path. There are three forces acting on the ball during its flight; gravity, air

resistance, and the Magnus force which causes the ball to curve. The force due to gravity

(mg) is always pointed straight down toward the Earth. Air resistance slows the ball, and

in the range of speeds encountered in tennis, the force it causes is proportional to the

square of the ball?s speed. For example, a ball moving at 50 m.p.h. will encounter four

times as much air resistance force than that of a ball moving at 20 m.p.h. Wind also

creates an air resistance force, which can be analyzed in a similar manner. Because air

resistance force is proportional to the square of the speed, a crosswind of 20 m.p.h. will

exert four times as much force on the ball as a 10 m.p.h. crosswind, and a 30 m.p.h.

crosswind provides a force nine times as strong as the 10 m.p.h. wind. This is obvious

when a tennis player tosses the ball up for a serve if there is a brisk breeze. The Magnus

force is at right angles to the direction that the ball is moving and is proportional to how

fast the ball is spinning. It is also proportional to the square of the ball?s speed. Because

of these factors, it is very important for tennis players to be able to observe these certain

characteristics. They must be able to think critically to place the shot in the correct side

of the opponent?s court.

There are many ways in which a player may hit the tennis ball. Choosing a good

strategy and position, hitting high-percentage shots, and using the proper equipment may

help the player win more points. The angle of the racket face and the direction of the

racket velocity at the instant of contact between the ball and the racket determine where

exactly the ball will go. When a player stands at the forehand corner of the court and

attempts to return a shot to the center of the challenger?s court with a forehand drive, the

shot will go crosscourt if the player swings a little early. If he or she swings a little late,

the shot will go down the line (Cantin 6). The swing of a tennis racket can be described

as the arc of a circle. At the second that the player hits the ball, the racket is in a certain

position in the arc. Thus, the face of the racket is pointing in a certain direction, and at

that moment the racket is moving tangent to the arc. The angular error of the racket is

given by the formula 57 x timing error x (ball speed + racket speed)/ swing radius. This

means that the worse the timing error, the larger the angular error. This error decreases

as the swing radius increases, but it increases as the racket speed and the speed of the

approaching ball increase. This attributes to the knots in a tennis player?s stomach as the

opponent puts increased pressure on them. Increasing the radius of swing however, will

improve the player?s accuracy and control. If the player keeps a firm wrist and uses his

or her shoulders as the pivot point for his or her shots, he or she will double the radius of

his or her swing and will reduce by half the horizontal angular error caused by the timing

error associated with that shot (Brody 119).

The three most popular techniques in the sport of tennis include topspin,

backspin, and sidespin. Topspin is, by far, the most challenging and requires a greater

appreciation of physics. Topspin on a tennis ball is usually called the powerspin. The

difference between a shot with topspin and a shot without topspin is rotational motion on

the shot with topspin as well as translational motion. If the face of the racket is oriented

so that it is perpendicular to the direction of the racket?s motion, the resulting shot will

have little or no spin. So how do you generate a lift and spin on the tennis ball? Lift is

generated by creating a pressure difference and deflecting the flow. To create a pressure

difference on the ball, it needs to move more fluid around one side than the other.

Spinning the ball will set up the imbalance, thus making the pressure difference. When

the tennis ball rotates, the fluid that is in contact with the ball?s surface tends to rotate

with the ball. The air next to the air on the surface tends to do the same thing. Far from

the ball, this rotation does not affect the surrounding air. Very close to the ball, however,

these fluid layers make up what is called a boundary layer. Consider the topspin stroke;

if the ball doesn?t rotate as it flies through the air, then both the top and bottom sides of

the ball meet the air rushing over it at the same speed. Relative to the ball, the top of the

ball in topspin spins forward into the oncoming air. There is more movement of air

towards the bottom surface. Now, more fluid needs to pass through the same space on

the underside of the ball. Basically, the flow is squashed on the lower side of the ball.

This means that there needs to be a higher velocity on the lower side of the ball, and,

subsequently, a lower velocity on the top of the ball. On the top side of the ball this

lower velocity creates a higher pressure. This effect is known as Bernoulli?s Law. With

high pressure on one side and low pressure on the other, there is an imbalance in the

forces on the ball. In the case of topspin, the higher pressure on the top curves the ball

downward from its straight line path.

Finally, to execute full understanding of topspins, one must be able to identify

rotational momentum and how it differs from other shots in tennis. Rotational motion is

the spinning of the ball as it sails across the net. Pure rotational motion describes the

principle that all points in the ball move in circles, and that the centers of these circles all

lie on a line called the axis of rotation. Because each point rotating with the ball has a

different linear velocity, spinning causes more air to flow over the top of the ball and thus

the ball falls shorter. If an object has points on it spinning, it has an access of rotation

which is located in the center of the ball.

Backspin and sidespin are also two other techniques in tennis, however, they are

not as interesting or as challenging as the topspin. Backspin is accomplished by

chopping at the ball with an upward tilt of the racket. The ball will be moving up, and

will remain high. The backspin shot floats the longest, and bounces very close to the

baseline. Thus, by successfully executing a backspin, a player reduces the margin for

allowable error (Bloom 2). Sidespin is yet another popular technique in the game of

tennis. Sidespin on a tennis ball makes the ball appear to be moving to the left or right.

Not only will the tennis ball look like it?s moving to the right or left, but it will remain

low when crossing the net. Spin is applied to the ball by the friction between the ball and

the strings when the ball slides or rolls across the racket face. The distance that the ball

slides or rolls across the racket is determined by the dwell time and the velocity of the

racket in the direction parallel to the racket face (Randall).

The height to which the ball bounces and the speed of the court are also subject to

those same laws. Tennis courts are made of all types of surfaces: clay, grass, concrete,

asphalt, and rubber. When a ball bounces on the court, its horizontal speed is reduced by

its interaction with the court?s surface. If the ball slows down a great deal upon

bouncing, the court is slow, while a fast court does not affect the ball?s horizontal speed

as much. There are two characteristics of a court surface that influence the ball as it

bounces. These characteristics are the coefficient of restitution and the coefficient of

friction between the ball and the surface. The coefficient of restitution determines how

high the ball will bounce from a certain height. It is defined as the ?ratio of vertical ball

speed after the bounce to the vertical ball speed before the bounce? (Brody 62). A high

coefficient of friction is a measure of the frictional force of the sort of surface on the

tennis ball in a direction parallel to the surface; it usually slows a ball down (See figure

1). A high value of the coefficient of friction means that the frictional force on the ball is

large. While coefficient of restitution influences the vertical velocity of the ball, the

friction affects the horizontal velocity of the ball, and that is the direction that determines

a court?s speed (Brody 63). The larger the friction between the ball, the more the ball

will slow down when it bounces, and the slower the court will be. When a ball with no

spin hits a court surface, there is a frictional force parallel to the surface and in a

direction opposite to the ball?s direction of motion. The ball will begin to slide or skid

along the court, with the bottom of the ball slowing down more than the rest of the ball;

this will cause the ball to rotate. If the frictional force is powerful enough and the ball?s

incident angle of bounce is large enough, the ball will begin to roll on the court surface

before it rebounds and loses contact with the ground. If the ball leaves the court before

rolling begins, it is considered to be a fast court. Aging of the court also determines the

speed of a court. Many hard courts must be resurfaced if the slowness that they have

when they are new is to be retained. These courts are covered with a latex that contains

sand. The roughness of the sand creates a great deal of friction between the surface and

the ball. As the court is played on, however, constant wear tends to smooth the surface,

reducing the friction. As a result, the court speeds up with age and use.

After gaining an understanding for the motion of the ball and the many forces it

encounters while in the air and on the court, it is important to understand the general

?anatomy? of a tennis racket and how to use its features to fully benefit a one?s game.

Most of tennis racket science is involved with technological improvements of the rackets

in order to improve performance on the court, much like my Radical Tour 260. Changes

in the racket have included composition of frames, string pattering, vibration-dampening

systems, and the overall head size. Wooden rackets were originally used until the early

1980s when it was discovered that graphite produced stiffer rackets, thus increasing the

power. Moreover, the enlargement of the head has been the most beneficial in terms of

performance. The basis of increasing the head size was to enlarge the sweet spot, the

precise area on the racket face that delivers the most powerful shot with the least amount

of vibration. Experiments by racket maker Howard Head, the developer of the idea of

larger heads for graphite rackets, revealed that ?increasing the face size by twenty percent

increased the sweet spot by nearly three hundred percent? (Brody 213).

A very practical question to ask a tennis player is what is the ideal racket? This is

the same question I asked myself and my teammates as I decided that the Radical Tour

260 was the racket for me. One must be aware of the principles of physics that go into

designing a high performance racket. These principles include the characteristics of

strings, center of percussion, racket vibrations, and moments of inertia.

The strings of a tennis racket play an important role in how the ball is hit. There

is more to strings than just tension. Years ago, when rackets were strung, the head sizes

were all the same and thus, the tension was also. Now, with a various head-sizes, a

tension of 65 pounds in a standard racket plays tightly, while 65 pounds in an oversize

frame may play too loosely. The way the racket plays with respect to the stings can

determine how much of the string plane deforms when a force is applied to the racket.

Rackets will play in a similar manner if they are strung so that their curves of string plane

deformation versus force are similar. By measuring the string plane deformation, I can

compare the Radical Tour 260 with a Wilson Kramer strung with 16- gauge string and

know how the strings in one will play in relation to the other (Brody 6). Also, if one

increases the tension of the strings in proportion to changes in the length of the strings in

the head, the string plane deformation is similar to the first. Simplistically stated, in

order to change from one frame size to another while retaining similar playing

characteristics from the strings, the tension divided by string length must be kept the

same. This is why the oversize racket is strung at higher tensions. One of the many

reasons that tennis uses rackets instead of paddles is so that the player can get power.

The goal is for the ball to leave the strings with a high velocity without having to swing

the racket. The tighter the racket is strung, the more it feels like a wooden board and the

less power the player will get. Why do loose strings give more power than tighter

strings? Tennis balls do not store and return energy efficiently. For example, imagine

throwing a tennis ball from a height of 100 inches onto a hard floor. The tennis ball only

rebounds to a height of about 55 inches, a loss of about 45 percent of the initial energy

of the ball. Strings, however, are designed to return 92.5 percent of the energy that is fed

to them (Watts 84). To give the ball the maximum energy, the strings must store the

energy by deflecting. If the strings have a lower tension, they will deflect more and the

ball will deform less. So why not string all rackets loosely? By reducing the tension too

much, the speed of the ball will be inadequate and the strings will wear out too fast from

excessive rubbing. Moreover, by stringing a racket loosely, control must be sacrificed.

Reasons for loss of control because of loose stringing includes: making the speed of the

ball more dependent upon the pace of the opponent?s shot, changing the angle at which

the ball leaves the racket, and increasing the dwell time of the ball on the strings. This

allows the racket to twist or turn more while the ball is still in contact. The looser the

strings, the longer the ball will reside on the strings. The dwell time of the ball on the

strings should increase as the inverse of the square root of the tension. In addition, the

dwell time of the ball on the strings decreases the harder the ball is hit, because the

strings become effectively stiffer the more they are forced to deform (Brody 12).

When a player hits a shot and feels great, he or she has hit the sweet spot.

According to the American Journal of Physics, there are three sweet spots of a racket

(Bloom 4). Sweet spot number one is the initial shock to a players hand. To some this is

known as finding the node of the first harmonic (See figure 3). Sweet spot number two is

when that uncomfortable vibration that many players feel is also a minimum. Sweet spot

number three is when the ball rebounds from the strings with maximum speed and

power. When a racket is struck by a ball, the racket recoils to conserve momentum. If

the ball hits the racket at its center of mass, the racket recoil is pure translation and there

would be no rotation of the racket. Instead, if the ball hits in the center of the strung

area, the racket both translates and rotates. If the ball is not hit exactly at a sweet spot,

however, there will be an initial net force on the player?s hand. If a player hits the ball

closer to his or her hand than this sweet spot, the initial force will pouch on the palm of

his or her hand.

The oscillation amplitude of the racket depends on the point of impact for the

occurring vibrations. When a racket hits the ball, the racket deforms due to the impact

and then begins to oscillate for tenths of seconds (See attachment 4 &5). Since most

tennis players, like myself are not able to hit the ball at the second sweet spot every time,

manufacturers have attempted to reduce the vibrations with special vibration-damping

materials. Some say these small devices that fit on the strings are purely psychological.

Research, however, shows that the feedback from the racket is dramatically affected.

These small devices ?damp the vibrations of the strings that oscillate up to 500 to 600

cycles per second? (Randall). In doing this, they change the sound of the interaction

between the ball and the racket.

When a tennis player hits the ball off-center, the racket tends to twist and the shot

is more than likely to go out of bounds. The property of the racket to resist this change in

twisting is known as the roll moment of inertia. The quantity m(r squared) represents the

rotational inertia of the particle and is called its moment of inertia. It is calculated as the

mass of the object times the distance of that mass from the axis squared. If the moment

of inertia is made larger, the racket is less likely to twist and will gain stability along the

long axis (Brody 214) (See figure 2). The moment of inertia can be increased by adding

masses along the outside edge of the head. The Wilson?s Hammer System was created to

do just this. The theory behind the Hammer (another racket) is ?that it is head heavy,

providing more power due to an increased moment of inertia? (Brody 214). In addition

to the head?s weight, the moment can be increased by increasing head-width. Because

inertia depends on the factor m(r squared), increasing the width also increases the polar

moment significantly more than increasing the mass. The polar movement is the

property of an object to resist twisting. Increasing the head on the racket reduces the

likelihood that the racket will twist in the player?s hand after an off center hit.

Through the understanding of the motion of the ball, characteristics of swings,

and general anatomy of the racket, one can see how physics influences even the most

basic aspects of tennis. Even though people participating in the game of tennis are not

completely aware of the physics in each shot, they are still able to enjoy the game. A

person who is seriously interested in the game of tennis, however, can figure out a lot by

studying the various laws of physics and how they determine the course of the sport of

tennis. That was my father?s intention when challenging me to research the Radical Tour

260. I did eventually obtain the racket. Through research? No, the coach called and

suggested the racket to my parents. Researching racket science and characteristics of the

sport of tennis has brought much humor to my parents. Was it fate that determined that I

would one day be researching the physics of tennis, or is this all a big dangerous

conspiracy between my professors, coaches, and parents?

Works Cited

Barnaby, John M. Racket Work- the Key to Tennis, Allyn and Bacon. Boston, MA. 1969.

Bloom, Phil. ?Finding Sweet Spots.? Phil Bloom.

(14 March 1998).

Brody, Howard. ?The Moment of Inertia of a Tennis Racket? Physics Today. April,

1985; (p. 213-215).

Brody, Howard. Tennis Science for Tennis Players, University of Pennsylvania Press.

Philadelphia, PA. 1987.

Cantin, Eugene. Topspin to Better Tennis, World Publications. Mountain View, CA.

1977.

Randall, James. ?The Tennis Racket,? Newton at the Bat: the Science in Sports. ed.

Schier and Allman. 1984.

Watts and Bahilli. Keeping Your Eye on the Ball, University of Pennsylvania Press.

Philadelphia, PA. 1994.