The Serve Calculations

The physics of volleyball behind this analysis is of a kinematic nature, since we are only concerned with the motion of the ball. This problem is an interesting application of projectile motion. To simplify this physics of volleyball analysis we shall assume that air resistance and aerodynamic effects acting on the volleyball can be ignored.

From the equations for projectile motion, we have

Where x and y denotes the position of the ball at any point in its trajectory, and t is time, in seconds.

Combine equations (1) and (2) to remove the time variable t and we get

This is the equation of a parabola in terms of x and y. This equation has the general form:

Where:

The variables a and b can be solved for in terms of the parameters L_a, L_b,h_o, and H. They can be solved using two simultaneous equations based on the coordinates of points B and C, relative to the coordinate system xy(with origin at point A).

The coordinates of point B (relative to xy) is (L_a , H—h_o)

The coordinates of point C (relative to xy) is (L_a+L_b , -h_o)

Where:

The coordinates (x,y) for points B and C can be substituted for x and y in the general parabola equation given by

We can then solve for a and b in terms of the parameters L_a, L_b, h_o, andH. These can then be used to solve for the initial velocity V and initial angle θ of the volleyball using the following equations:

The time that the ball is airborne (i.e. the time we wish to minimize) is given by

Upon analysis of the results we find that we can minimize the time by doing three things:

(1) Get the ball just over the net

(2) Make L_b as large as possible (serve the ball so that it lands near the end line)

(3) Make h_o as large as possible (with a jump serve)

Points (1) and (2) make sense since a shallower trajectory means the ball reaches a lower maximum height h_max, which means the ball spends less time in the air. However, if the ball lands close to the net (with small L_b), then the ball requires a high arc. This means that the ball is airborne for a longer period of time.

Point (3) makes sense since serving the ball at an h_o as large as possible (with a jump serve), enables the ball to start its downward trajectory sooner (since h_max is reached sooner). This also decreases the time the ball spends in the air.

To get an idea of how much time the ball spends in the air, let's say we have d = 9 m, h_o = 3.0 m, and H = 2.4 m. The time the ball spends in the air is t = 0.86 seconds.

A volleyball player can put the above three points into practice by practicing jump serves which (1) barely get the ball over the net, and (2) land as close as possible to the end line. The picture below shows an example of a jump serve.

In addition, serving the ball at a cross-court angle α does not change the time the ball spends in the air (for a given d, h_o, and H). It only affects the horizontal speed of the ball (Vcosθ). So, the greater the angle α, the greater the ball speed. This can be advantageous since a higher serve velocity V can make it more difficult for the opposing team to return the shot.