
Tennis Matrix
Exploring the Physics Behind Tennis

Racquets
Work and Power
Work = the energy used to generate a certain ball speed.
Work also measures how energy-efficient the racquet is.
A racquet that has low work is good and a racquet with high work
isn't.
Why?
A racquet with high work will require a player to swing and the hit the ball much harder in order to achieve the same ball speed and power as a low work racquet. Work can also be defined as a measure of the racquet's power. So, the less work the player has to generate, the more powerful the racquet.
In the game of tennis, power and work are inversely related. Low work means that the racquet is high in power while high work means the racquet is low in power.
Head heavy racquets tend to require a lot more work to hit the ball fast, which causes a strain on the wrist, arms and shoulder.
The Ideal Racquet
A heavy head, light racquet with a high swing weight is the most ideal work/power racquet. It is efficient and gives the user the best work for power output, which also leads to better control.
Control
Some people believe that a large head size leads to better control simply because there is more surface area therefore, it has a bigger sweet spot and the ball will bounce more accurately. This is somewhat true, however, a larger head causes the strings to deform more if the ball is hit off centre which will cause the balls to go all over the place and not necessarily where they want it to go.
Ideal Racquet
A mid-sized head with a handle-heavy setting is
ideal for overall control.
Vibration and Shock
In tennis, when the ball hits the racquet, the racquet bends. The elastic force works to restore the bent racquet to its original position and in doing so the racquet build up momentum and overshoots its original position. This results in the racquet going back and forth until the racquet loses energy through internal friction. This is known as Vibration.
Vibration is a wonderful indicator of dynamic racquet stiffness. When the ball impacts the racquet it is an instantaneous event and the amount of racquet deflection depends on its stiffness (elastic restorative force) and mass distribution.
Stiffness is defined as the resistance to bending. Therefore, bending is directly proportional to the degree of stiffness and bending depends on mass because a heavy mass means that it is more difficult to accelerate into a bending motion.
The combined influence of stiffness and mass is measured by the racquet's vibration frequency.
Frequency is number of back-and-forth cycles that the racquet completes in one second which is measured in Hertz (Hz), where 1 Hz = 1 cycle per second.
Frequency = 1/period.
The period is known as the time is takes one cycle of the vibration to be completed.
Period = time/# of cycles
The maximum distance a point on the racquet travels back-and-forth from its equilibrium position is the amplitude of the vibration.
Amplitude = (max-min)/2
The stiffer and lighter the racquet, the faster the frequency is, the shorter the period is, and the less the amplitude it has. And the softer and heavier the racquet, the slower the frequency, the longer the period and the greater the amplitude.
The vibration frequency stays the same no matter where the ball strikes on the string surface with the exception of vibration node, or vibration sweetspot. If you hit this point, the racquet will not bend. This is explained in the section Strings and Balls.
Analyzing Power and Racquet Speed
A racquet's power potential is determined by measuring ball velocities and calculating the ratio of the ball's outgoing bounce speed to its impact speed. The impact speed is defined as the combined speed of the racquet and ball just prior to collision.
Impact speed = ball speed + racquet speed
The power potential is always a fixed percentage for a given hitting location on the racquet. The impact speed is a different percentage for each location, and generally varies from racquet to racquet.
For example, if the power potential is 40% in the center of the racquet, then the ball will bounce from the racquet at 40% of the impact speed.
Bounce speed = power potential x impact speed
The speed of this bounce is determined by frame stiffness, and string-bed stiffness at the impact location.
However, bounce speed is only one component of the final shot speed. The other component is the racquet speed. The bounce occurs off a racquet that is already traveling at a given speed. The bounce speed is added to the speed of the racquet from which it bounces to get the shot speed:
Shot speed = bounce speed + racquet speed

The graph to the side shows a measurement of the acceleration of a racquet's handle at a location 10 cm from the butt end resulting from an impact at 25 inches from the handle butt. Itshows the acceleration of this spot as it changes its motion back and forth.
What the tennis player feels when they hit the ball is is shock and vibration in their hand. Shock can be defined as the difference in kinetic energy after impact. This generates vibrations in the frame. The change in kinetic energy refers to how a racquet is decelerated when it makes contact with the ball.
Energy is used before impact. Energy is used to speed up the racquet for the swing, and more energy is expended when the ball is controlled during the swing. The tennis ball receives part of the lost energy while the rest of the energy then becomes internal energy, wasted and absorbed by the bending of the frame. It is important that the frame is not too stiff and light because the energy that usually gets absorbed by the frame's bending will end up being absorbed by the arm holding on to the racquet.

