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?
Barnaby, John M. Racket Work- the Key to Tennis, Allyn and Bacon. Boston, MA. 1969.
Bloom, Phil. "Finding Sweet Spots." Phil Bloom.
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.
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.