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One way to optimize a volleyball serve is to minimize the time the ball spends in the air. This in turn minimizes the reaction time of the opposing team, making it more difficult for them to return the shot. In this analysis of the volleyball physics, we will look at ways to minimize the time the ball spends in the air, after the serve is made.

To set up this physics analysis we must first define the different variables in the problem. The schematic below shows a top view of a volleyball court, with labels given as shown.

Where:

The following schematic shows a view of the volleyball trajectory, between the point of serve and the point at which the volleyball lands on the court.

Where:

Point

Point

Point

The coordinate system

The physics behind this analysis is of a kinematic nature, since we are only concerned with the motion of the ball. This optimization problem is an interesting application of projectile motion. To simplify this 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

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

This is the equation of a parabola in terms of

Where:

The variables

The coordinates of point

The coordinates of point

Where:

The coordinates (

We can then solve for

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

(3) Make

Points (1) and (2) make sense since a shallower trajectory means the ball reaches a lower maximum height

Point (3) makes sense since serving the ball at an

To get an idea of how much time the ball spends in the air, let's say we have

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.

Source: http://en.wikipedia.org/wiki/Volleyball. Author: http://commons.wikimedia.org/wiki/User:Spangineer

In addition, serving the ball at a cross-court angle

The analysis shown previously allows us to predict the primary kinematic behaviour of a volleyball serve, subject to the assumption that air drag and aerodynamic effects can be ignored. However, these effects can in fact be significant and must be accounted for in order to make the model prediction as accurate as possible. In the next section we will discuss these effects in greater detail.

The airborne time of the volleyball can be reduced even more by putting top-spin on the volleyball. This causes the ball to experience an aerodynamic force known as the magnus effect, which "pushes" the ball downward so that it lands faster. This complicates the physics analysis. The figure below illustrates the magnus effect.

Source: http://en.wikipedia.org/wiki/Magnus_effect. Author: http://en.wikipedia.org/wiki/User:Gang65

As the ball spins, friction between the ball and air causes the air to react to the direction of spin of the ball.

As the ball undergoes top-spin (shown as clockwise rotation in the figure), it causes the velocity of the air around the top half of the ball to become less than the air velocity around the bottom half of the ball. This is because the tangential velocity of the ball in the top half acts in the opposite direction to the airflow, and the tangential velocity of the ball in the bottom half acts in the same direction as the airflow. In the figure shown, the airflow is in the leftward direction, relative to the ball.

Since the (resultant) air speed around the top half of the ball is less than the air speed around the bottom half of the ball, the pressure is greater on the top of the ball. This causes a net downward force (F) to act on the ball. This is due to Bernoulli's principle which states that when air velocity decreases, air pressure increases (and vice-versa).

As a result, by putting enough top-spin on the volleyball, the airborne time can be further reduced by as much as a tenth of a second. The following paper by D. Lithio and E. Webb explains the details of a volleyball serve and include mathematical models which account for the effects of air resistance (drag) and top-spin:

This paper is very informative for those wishing to see the full analysis of the physics of volleyball, with regards to optimizing the serve.

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