The most
fundamental part of the game of tennis is the rally,
where opponents successively hit the ball back and
forth across the net, using their tennis racquets,
until one player makes an error. During a rally,
there is an amazing array of basic mechanical
principles underway that govern the trajectory or
the ball.

The dynamics of motion of a
tennis ball are governed by the same basic
mechanical principles that we have studied in
General Physics I. However, the sheer number of
force interactions that occur on a tennis ball make
it impossible to easily derive a simple analytical
equation for the flight path of a tennis ball once
it has been hit by a racquet. So instead we will
briefly touch on some of the physical principles
that govern the movement of a tennis ball. Then we
will examine the effects of spin on the flight and
bounce of a tennis ball.

Newton’s Three Laws of Motion:

Newton’s three laws of motion are the underlying
physical principles that govern the movement of
objects, including tennis balls! The number of ways
that Newton's laws of motion apply to the movement
of tennis balls are too numerous to list here, but a
few demonstrative example will be given. Whenever a
tennis racquet hits a ball, Newton's Second
Law, $\backslash vecF=m\backslash veca$,
which relates the external force on object to the
object's resulting acceleration, determines the
resulting acceleration of the ball due to the force
of the racquet on the ball. In addition, Newton's
Third Law informs us that that the force exerted by
the ball on the racquet is equal in magnitude and
opposite in direction to the force exerted on the
racquet by the ball.

Gravity:

Gravity is what
allows the game of tennis to be played. Without
gravity, the ball wouldn't come down and bounce in
the court after it was hit. A tennis ball is
constantly undergoing an acceleration due to the
force of gravity. The force of gravity acts downward
on the ball, and is perpendicular to the court
surface. An equation for the magnitude of the force
of gravity on the ball is given by: $$F\_g=mg$$where
m is the mass of the ball, approximately 57 g
by regulation (Source: [5]), and g is
the acceleration due to gravity near the surface of
the earth, $g=9.81\backslash frac\{\backslash text\{m\}\}\{\backslash text\{s\}^2\}$.
Thus there is a constant downward force of
$F\_g=(0.057\backslash text\{kg\})(9.81\backslash frac\{\backslash text\{m\}\}\{\backslash text\{s\}^2\})=0.56\backslash text\{N\}$
on the ball at all times.
$F\_g=mg$

Impulse and Momentum:

Img Src: [3]

When a tennis
racquet hits a tennis ball, the force of the racquet
on the ball delivers an impulse to the ball while
the ball is in contact with the racquet. The
magnitude of the force of the racquet on the ball
varies with time, starting low at initial contact,
then reaching a maximum when the ball compression
and racquet string deformity reaches a maximum,
before reducing back to zero as the ball leaves the
racquet strings. The Impulse-Momentum Theorem states
that $\backslash vecJ=\backslash vecp\_f-\backslash vecp\_i=\backslash Delta\; \backslash vecp$, where J
is the impulse on the ball and p is the
momentum of the ball. That is, the impulse on the
ball is equal to change in momentum of the ball.
Figure (a) below shows the general shape of the
force curve of the racquet on the ball over time
during the impact. The magnitude of the impulse is
given by the area under the F vs. t
curve.

Source: [3]

Img Src: [3]

Collision Elasticity:

A tennis
racquet's strings are highly elastic and thus exert
a strong restoring force on a ball when it impacts
the strings, which helps add to the impulse
delivered to a tennis ball when it is hit. A tennis
ball is relatively inelastic compared to a the
strings of a tennis racquet. As a result, tennis
balls lose energy to dissipative forces when they
bounce due to friction between the ball and the
court and deforming of the tennis ball itself. One
way of quantifying the elasticity of an object, such
as a tennis ball, is through measuring its
coefficient of restitution on a surface, such as a
tennis court. We discuss coefficients of restitution
further in a later section when we examine the
bounce of a tennis ball.

When
a ball flies through the air, it experiences a drag
force that acts in the direction opposite of the
motion of the ball. An equation for the
magnitude of the drag force on the tennis ball is
given by: $$F\_d=\backslash frac\{1\}\{2\}C\backslash rhoAv^2$$where
$C$ is the drag
coefficient,
$\backslash rho$
is the density of air,
$A$
is cross-section area of the object, and
$v$
is the magnitude of the velocity of the object. The
magnitude of the force of drag on the ball is
directly proportional to the square of the magnitude
of the velocity of the ball, so the drag force
increases as the ball speeds up. Drag slows down
tennis balls as they fly through the air.

Friction of the
strings of a racquet on a tennis ball is what allows
spin to be imparted on the ball during a hit.
Furthermore, friction between the ball and the court
affects the way that the ball bounces. Friction
force on an object is proportional to the normal
force on the object and can be calculated by $f.$where $f$
is the magnitude of the friction force,
$n$ is the
magnitude of the normal force, and $\backslash mu$
is the coefficient of kinetic friction between the
object and the surface. Kinetic friction is
perpendicular to the normal force and opposite in
direction to the velocity vector.

Source: [3]

Rotational Motion, Center of
Mass, and Moment of Inertia:

Rotational
motion plays a huge role in the game of tennis. The
rotational motion, or spin, of a tennis ball affects
its trajectory through the air, as well as how it
bounces. Players make constant use ball spin to
affect their shots and make them more difficult to
return. We will spend the next two sections
exploring the effects of ball spin on the flight and
bounce of a tennis ball.
The center of mass of a tennis
ball is at the center of a tennis ball since it is a
spherical shell with uniform density. The moment of
inertia about the center of a tennis ball can be
calculated using the formula for the moment of
inertia about the center of mass of a uniform
spherical shell:$$I=\backslash frac\{2m\}\{5\}\backslash left[\backslash frac\{r\_2^5-r\_1^5\}\{r\_2^3-r\_1^3\}\backslash righ$$
where $m$
is the mass of the ball,
$r\_1$
is the inner radius, and
$r\_2$
is the outer radius of the ball. Since our analysis
will be primarily qualitative, we will not calculate
the precise moment of inertia here. Because tennis
balls are hollow, and their mass is concentrated far
from the center of the ball, they have a larger
moment of inertia and will be more resistant to
rotation relative to uniform density solid balls
with the same radius and mass.
The torque about the center of
the ball caused by friction forces imparts a
rotational acceleration on the ball, since
$\backslash tau\; =\; I\backslash alpha$,
where
$\backslash tau$ is the torque,
$I$
is the moment of inertia about the center of the
ball, and
$\backslash alpha$
is the rotational acceleration that results. This
torque from friction is the primary way that tennis
balls get their spin.