The
bounce of a tennis ball greatly affects the
way in which the receiver can return it. Much
as the spin plays a large role in the
trajectory of a ball, it also strongly affects
how the ball bounces. Knowing how spin affects
a ball's bounce is a key part of the game of
tennis.
In order to analyze the
motion of a tennis ball before, during, and
after its bounce, its movement must be broken
down into separate horizontal and vertical
components. In addition, for our purposes we
will continue to identify rotational motion of
a tennis ball in the direction of topspin as
forward rotation/spin, and rotational motion
in the direction of underspin as backward
rotation/spin. The change in the vertical and
horizontal components of a tennis ball's
velocity during the bounce are different for
each type of shot: flat, topspin, and
underspin. It is the ratio of the ball's
vertical to horizontal speeds after the bounce
that determines the angle at which the ball
bounces off the court.
There are two physical
properties that are highly important in
understanding the dynamics of a tennis ball
bounce. The first of these is the coefficient
of restitution, or COR, of the ball on the
court. The COR is determined by dropping a
ball onto the court, and it is the ratio of
the relative speed of of the ball just after
colliding with the court to its relative speed
just before colliding with the court. The COR
affects a ball's vertical speed after its
bounce. The second important property is the
coefficient of kinetic friction, or COF,
between the court and the ball. The force of
friction of the court on the ball affects the
balls spin and horizontal speed, but plays not
role in determining its final vertical speed
after the bounce. For all tennis courts, the
COR and COF of a tennis ball on the court are
between 0.5 and 1.0.
Even though a tennis ball
is only in contact with the ground for between
5 and 7 milliseconds, the motion dynamics of
the ball change dramatically during that time.
The explanations given below are not an
exhaustive account of the full dynamics of a
bouncing tennis ball, but are instead meant as
a survey of the underlying bounce properties
of tennis balls with varying spins.
Source: [4]
Spin Change:
When a
tennis ball impacts the court, it initially
skids and the court exerts a kinetic friction
force on the bottom of the ball in the
direction opposite that of the ball's
horizontal motion (except in a special case of
topspin, which is discussed below). This
friction force exerts a torque about the
center of the ball that causes forward
rotational acceleration on the ball. This
results in the ball taking on topspin as it
leaves the bounce, if it did not already have
some. For balls hit with underspin, this
results in a complete reversal in direction of
the spin of the ball during the bounce. Balls
that already have topspin will generally have
their topspin increased during the bounce, but
it can also be decreased depending on the rate
of rotation of the ball. This will be further
elaborated on in the Topspin section.
Source: [4]
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Flat shot:
A tennis
shot hit with no spin, or a flat shot, will
very nearly reflect off the surface of the
court during its bounce. This means that the
angle at which the ball impacts the ground,
the angle of incident, is approximately equal
to angle at which it leaves the ground, the
angle of reflection. The ball loses vertical
speed because the COR less than 1.0. In
addition, the ball loses horizontal speed
because the friction force of the court on the
ball acts in the direction opposite of the
horizontal movement of the tennis ball, and
the COF of the ball on the court is greater
than 0. This results in the ratio of the
vertical speed to the horizontal speed of the
ball before and after the bounce being very
close in value. Hence, the ball leaves the
bounce with approximately the same angle as
its angle of incident, but with a slower speed
due to smaller horizontal and vertical
velocity components. As discussed earlier, the
ball have some topspin after the bounce.
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Source: [4]
Topspin:
Img Src:
http://www.tennisindustrymag.com/issues/200404/images/200404ball_bounce.png
When a
ball hit with topspin impacts the ground, the
forward spin of the ball reduces the effect of
the friction force of the court on the ball.
If the tangential velocity of the top of the
ball is less than the horizontal velocity of
the ball, then while the ball is in contact
with the court during the bounce, the force of
friction increases the ball's forward angular
velocity. Once the tangential velocity of the
top of the ball is the same as the overall
horizontal velocity of the ball, the ball
begins to roll, which reduces the friction of
the court on the ball dramatically. This
causes a smaller net friction force to be
applied to a ball with topspin during its
bounce and results in a ball losing less
horizontal speed. The smaller decrease in
horizontal speed of the ball causes the ratio
of vertical speed to horizontal speed to be
lower after the bounce. This results in the
ball to leaving the ground at an angle lower
than its angle of incident. Further, since
both the vertical and horizontal speeds of the
ball are decreased during the bounce, the
overall speed of the ball is also decreased.
In the case that the
tangential velocity of the top of the ball is
the greater than the overall horizontal
velocity of the ball, the friction force of
the court on the ball will instead point in
the direction of the overall horizontal
velocity of the ball. This will reduce the
angular velocity the ball until the tangential
velocity of the top of the ball is the same as
the overall horizontal velocity of the ball,
at which point the ball will begin rolling and
the friction force of the court on the ball
will be greatly reduced. The force of friction
results in an increase in horizontal speed of
the ball. Thus the ratio of vertical speed to
horizontal speed of the ball is again lower
after the bounce. Therefore the ball still
leaves the ground at an angle lower than its
angle of incident.
Although the angle at which
topspin balls leave the bounce is less than
their angle of incident, people do not
perceive topspin bounces as such. Topspin
shots are often used to create a high bounce
that is difficult to return. Because balls hit
with topspin are usually hit higher over the
net than other shots and have a downward
Magnus force acting on them, they generally
impact the court with a greater vertical speed
than other shots, resulting in an overall
higher bounce. Thus, we perceive balls hit
with topspin to have a higher effective
bounce, despite the fact that the spin of the
ball reduces the angle at which the ball
leaves the ground.
Source: [4]
A Shot Hit With Topspin
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Underspin:
Img Src:
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When a ball with underspin
impacts the ground, the ball skids across the
surface of the court, and the court exerts a
kinetic friction force on the ball in direction
opposite that of the overall horizontal motion
of the ball. Since the ball has backward
rotation, the kinetic friction force must act
for a longer period of time on the ball before
it will start rotating forward. This increases
the net friction force on the ball and reduces
its horizontal speed by a greater amount than
flat and topspin shots. As a result the ratio of
vertical speed to horizontal speed of the ball
is higher after the bounce than before it. Thus
the ball leaves the ground with an angle greater
than its angle of incident. Further, since both
the vertical and horizontal speed of the ball
are decreased, the overall speed of the ball is
also decreased.
Similar to topspin shots, we
perceive the bounce of of an underspin shot
differently because of how they are used in
tennis. Although the angle that an underspin
shot leaves the ground with is greater than its
angle of incidence, underspin shots are often
thought of as having a low bounce. This is
because underspin shots are usually hit low
above the net and have upward Magnus force
acting on them, which causes them to impact the
court with a lower vertical speed than other
shots, resulting in an overall lower bounce.
Thus, we perceive balls hit with underspin to
have a lower effective bounce, despite the fact
that the spin of the ball increases the angle at
which the ball leaves the ground.
Source: [4]
A Shot Hit With Underspin
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