# Basic Tennis Physics

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, $\vecF=m\veca$, 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, $g=9.81\frac\left\{\text\left\{m\right\}\right\}\left\{\text\left\{s\right\}^2\right\}$. Thus there is a constant downward force of $F_g=\left(0.057\text\left\{kg\right\}\right)\left(9.81\frac\left\{\text\left\{m\right\}\right\}\left\{\text\left\{s\right\}^2\right\}\right)=0.56\text\left\{N\right\}$ 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 $\vecJ=\vecp_f-\vecp_i=\Delta \vecp$, 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.

Source: [7]

Img Src: http://deansomerset.com/wp-content/uploads/2011/11/tennis-ball-impact.jpg

### Drag:

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:
$F_d=\frac\left\{1\right\}\left\{2\right\}C\rhoAv^2$where is the drag coefficient, $\rho$ 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.

Source: [3]

Img Src: http://en.wikipedia.org/wiki/Trajectory_of_a_projectile

### Friction:

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 where $f$ is the magnitude of the friction force, is the magnitude of the normal force, and  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:$I=\frac\left\{2m\right\}\left\{5\right\}\left\left[\frac\left\{r_2^5-r_1^5\right\}\left\{r_2^3-r_1^3\right\}\righ$ 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 $\tau = I\alpha$, where is the torque, $I$ is the moment of inertia about the center of the ball, and $\alpha$ is the rotational acceleration that results. This torque from friction is the primary way that tennis balls get their spin.

Source: [3]