The Laws of Physics in Volleyball

Volleyball, like many other sports, has many factors that result in a large number of unexpected outcomes. Such phenomena may be given a rule and order using physics and mathematics, allowing them to be studied and recreated. The benefit of creating a rule is that it allows for performance optimization and prediction, making sterile statistics obsolete. Technical gestures are examined in volleyball in order to rectify them as much as possible and achieve maximum strength, power, and efficiency.


Technical gestures are an essential component of an athlete’s or a coach’s skill set. The efficacy of the movement is critical in every action, and a more efficient technical movement might be crucial for the match’s final score. In the past, video recordings of matches were the only way to examine motions. Today, thanks to computers, high-speed cameras, and, above all, advanced software, details can be analyzed in depth to improve the final results.

Physics is at the basis of all processes, and linear velocity, angular velocity, acceleration, and mass are at the basis of any mathematical computation. It is possible to improve the performance of athletes during a match by using computerized movement analysis (see Figure 1† We may therefore substitute an active analysis for a passive one. Our attention will be on the best movement, which will allow us to try to understand which factors have resulted in a certain impact. We will then work with the player to enhance his or her particular gesture. The collected data is compared and analyzed by complex algorithms, which results in the optimization of the particular movement.

Figure 1: The study of the best performances is based on sophisticated physical analysis

The volleyball serve

A volleyball serves immerses a body in a fluid. The distance between the start and the finish is covered by a trajectory that is determined by many elements such as the angle of the start, the direction, and the speed of the ball. With the serve, the ball is put back into play with each action. It’s the team’s first offensive attempt. It can be done in a variety of ways, with varying degrees of force and precision:

  • from below;
  • Floating stop from above;
  • From above while jumping.

Hitting the ball can be characterized by power and precision. By properly adjusting these two opposite parameters, the team can decide for different strategies to be adopted in the match. The position of the hands can be decisive. In fact, the serve is very different if the ball is hit with the palm of the hand or with the fist. As you can see in figure 2, the trajectory of the ball is different with various types of serve. The trajectory of the ball in the low serve is parabolic so it is a slow serve with a very predictable course. In the floating serve, the ball is hit so that it does not rotate around its own axis. In this way, the ball passes over the net at a non-constant and undulating speed and it will tend to brake and deviate from its trajectory, unpredictably. If the exit angle the ball is upward, the parabolic range is lengthened. If the angle is downward, it is shortened. Today, with technology, it is possible to use “ball-shooting” machines with which the most powerful shots can be simulated in any condition.

The strongest and most powerful shots can reach, on average, the speed of about 100 km/h. However, there have been athletes who far exceeded the speed of 130 km/h (36 m/s). A ball shot at that speed has a kinetic energy of 181 joules, according to the following formula:

Serve Conditions
Figure 2: The type of serve conditions, speed and trajectory of the ball

The average characteristics of the ball and the volleyball court are as follows:

  • circumference: between 65 and 67 cm;
  • weight: between 260 and 280 gr;
  • internal pressure: between 0.3 and 0.325 kgf/cm;
  • real or synthetic leather material;
  • spherical shape;
  • the color of the ball is white or striped (for example white, red and green);
  • length of the court: 18 meters;
  • width of the court: 9 meters;
  • height of the net in the center: 2.43 meters (for men) and 2.24 meters (for women).

In schools, teachers propose many problems of mathematical physics to the students that are related to volley shots. For example, such types of questions are interesting: “A volleyball player performs a serve by throwing the ball into the position of the baseline from a height of 2.30 m and with an initial velocity of module v0, the direction of which forms an alpha angle = 45° with respect to the horizon. Calculate the minimum speed to allow the ball cross the net”. With a physics simulation software, these problems are quite easy (see figure 3† The size of the ball is slightly increased in the drawing, to be clearly visible. In the simulation, the ball is thrown at five different initial speeds:

  • 5m/s;
  • 7m/s;
  • 9m/s;
  • 10m/s;
  • 12 m/s.
Figure 3: With simulation software it is quite easy to solve any type of problem
Figure 3: With simulation software, it is quite easy to solve any type of problem

The serve in the first and second cases is really weak and the ball does not reach the opponent’s court. The third bar touches the net and barely manages to cross the field. The fourth and fifth bars easily cross the net.

Today, there are sophisticated ball throwing robots that perform any type of stroke with great precision. They are made of metal and run on batteries, which makes it possible to take them anywhere. The safety is also extremely accurate. Indeed, it has sensors that allow to immediately stop the shooting of the ball if there is a person in the immediate vicinity. They make it possible to set different shooting directions and perform special effects on the ball. You can also choose the shooting cadence, from a few shots per minute up to 100 and more shots per minute. All professional volleyball teams train with these machines.

Deformation of the ball

ball serve
Figure 4: During the serve, the ball undergoes impressive temporary structural deformations

In volleyball, the phenomenon of the ball deformation occurs every time it collides with the hands of the athletes. The ball, in fact, undergoes a deformation due to the impulse generated on the surface. The deformation is clear when hitting the floor and in rebate at the moment of the blow. It can only be highlighted in a “still image” of a high-speed capture video, since it only lasts a few milliseconds. The ball returns to its original state during the parabolic trajectory and this happens, on average, before passing over the net. The restoration of the equilibrium condition occurs because the internal pressure, due to the air present inside, pushes in the same way in all directions, returning the ball to its original shape, given the elasticity of the components. When the ball comes into contact with another athlete again, it is back to its original shape and will undergo a new deformation after the new contact. As can be seen in figure 4, it is calculated that on average, during a strong hit, the ball deforms on the diameter of about 6 cm (3 cm on one side and 3 cm on the other side). It is temporarily reduced, from 21 cm to 15 cm.

Parabolic motion in volleyball

A volleyball serve can be assimilated to the parabolic motion with an oblique shot. The ball is thrown with “v0” speed with an “alpha” launch angle with respect to the horizontal. In a reference system whose origin “O” is the launch point of the ball, it falls back to the ground after having described a parabolic trajectory. The motion of the ball is, at any moment, the result of two different independent motions that take place in the x and y axes. It is subject to a downward acceleration of an intensity proportional to the acceleration of gravity and is equivalent to:

On the x axis, the motion is uniform rectilinear with a constant horizontal speed equal to v(0x). The hourly law is equal to:

On the y axis, the motion is uniformly decelerated rectilinear during the ascent (from O to V) and uniformly accelerated rectilinear during the descent (from V to A). The law is:

where v (0y) is the component of the initial speed along the y axis. The velocity components are calculated with the following relations:

We get the equation of time:

and the equation of a parabola (see figure 5

Also with the following two constants:

The parabolic equation is as follows:

The parabola has the vertex at point V and the concavity facing downwards.

parabolic trajectory
Figure 5: the parabolic trajectory of the ball

The “rise” of the ball, that is the time it takes to go from O to V, is characterized by a vertical with uniformly decelerated rectilinear motion (acceleration is equal to −g) with initial velocity equal to v(0y) and velocity final on the y axis equal to zero. It is equivalent to:

The flight time, ie the time it takes to travel the distance from O to A, is equal to double the climb time.

The determination of the maximum height of the ball is also very interesting and is equal to:

As can be seen from all the formulas and equations, the force of gravity plays a primary role in any direct and indirect movement. A volleyball match played on another planet would certainly be a different story and athletes would behave as if they had never played volleyball.

This article originally appeared on sister site EEWeb

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Physics in Basketball


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