Tennis can be a fascinating way to study and apply both simple and advanced physics concepts. Moments of inertia, elastic impacts, and momentum are all part of a tennis match. How can a physicist help a tennis player? The technical preparation of tennis players takes physics into account, and most training tactics focus on concepts such as point of impact, balance, inertia, and mass.
The racket (see figure 1) is usually six times heavier than the ball and around one-sixth the weight of the player’s arm in sports where a tool is used to hit a ball. The tennis ball is 57 grams in weight. An adult’s arm weighs about 2 kg. The ideal weight of a racket is about 340 grams. What is the significance of this numerical correlation? When the weight of the tool surpasses this ratio, the weight of the racket tends to slow down the arm’s speed. The speed with which the racket strikes the ball is also determined by its weight. A ball struck with a heavier instrument will go quicker at the same speed.
When two items meet, such as a racket and a ball, they generate a force that causes them to change their state of motion. According to the following formula, the force required to change a body’s state of motion, i.e. accelerate or decelerate it, is proportional to its mass:
where:
“F” is the force that modifies the state of motion;
“m” is the mass;
“a” is the acceleration.
To accelerate or decelerate a body with twice the mass of the other, you would require twice the force. When two moving bodies collide, a certain quantity of kinetic energy is released, which is determined by their mass and speed (kinetic energy is equal to half the mass times the speed squared). Because the ball and the strings are elastic, they are able to save a certain amount of kinetic energy when they collide, changing it into elastic energy that may then be returned to the bodies. Due to friction, a portion of the collected elastic potential energy is inevitably lost and turned into vibrations and heat.
The ball dissipates around 45% of its elastic energy when it collides with the strings. The rules of tennis demand this loss to prevent the ball from going too quickly and becoming harmful. In general, there is no such thing as a flawless racket, but a player should always try out many models before deciding on the one he or she will use in competitions.
Nowadays, to improve performance, tennis players study the different behaviors of any action. With the help of their trainers, players perform kinematic and dynamic analysis to study the relationships between the forces acting on a body and its motion. In tennis, the most studied gesture is the serve, because it is a shot performed with a series of precise movements and does not depend on the opponent or other agents of the game in progress (see figure 2). The serve is a very important shot because it starts the game but also offers the possibility of winning the point immediately. It is also a double-edged sword because, with two wrong serves, the player loses the point due to a double fault. The power of the shot is linked to the thrust of the legs, the rotation of the trunk, the action of the shoulders and the rotation of the arm around it, together with the position of the racket with a variable distance from the rotation axis, according to the formula:
This formula makes it possible to assert that the inertia of the body is all the greater as the distance to the axis of rotation is large.
Physics, together with mathematics, are the backbones of any sporting event. In a tennis match, Newton’s three laws can be observed at all times:
The principle of inertia states that a body remains in its state of rest or uniform rectilinear motion until an external force intervenes and modifies this state;
The second principle of dynamics states that a force acting on a body determines an acceleration which is directly proportional to the intensity of the force and inversely proportional to the mass of the body;
The principle of action and reaction states that, if a force acts on a body, there is another body on which an equal and opposite force acts.
What forces act on a tennis ball? The speed of the ball after impact with the racket strings is given by:
v (inp) is the speed of the incoming ball;
V (r) is the speed of the racket head at the point of impact;
and (A) is the apparent refund coefficient.
With a string tension of 280 N, the apparent restitution coefficient is about 0.4. If the string tension is reduced to 224 N, the apparent return coefficient increases by 7%, becoming 0.433. The result is a 3% increase in ball speed. A racket with a tension of 230 N produces, in practice, fewer errors, while a racket of 180 N produces more errors.
The sweet spot is a point on the racket strings that is stressed by the ball upon impact. This is where you have the best rebound on the stringbed. A tennis racket has two sweet spots. If the ball is hit in one of these two areas, the forces transmitted from the racket frame to the arm are extremely small and the vibrations are not detected by the player. These points cause a certain vibration of the strings around 100 Hz (for a flexible frame) and around 185 Hz (for a rigid frame). The impact of the ball on the racket is extremely short, only 5 milliseconds. The sweet spot does not coincide with the point where the ball receives the maximum thrust. A less rigid and more flexible racket produces less vibrations and is softer and less damaging to the joints. Furthermore, the vibration frequency of a racket depends on the stiffness of the frame (see figure 3). A rigid racket vibrates at 180Hz or more, while a flexible one at 140Hz or less. There is also another critical point, the “dead spot”, in which the energy is not returned entirely from the racket to the ball itself because the actual mass of the racket at that point is equal to the mass of the ball. The various points are placed in different places and are not fixed for every type of racquet. Raising the weight of the racket will lower or raise the positions of those points. One might think that the best spot on the racket to hit the ball with is the one at the center, but that’s not the case. Obviously, to prevent the handle from rotating in the hand, the impact must occur along the vertical axis as an extension of the handle. In the center of the strings, the kickback on the hand is minimal but a lot of power is lost due to the production of vibrations on the arm. The slightest vibration occurs when the blow occurs just above the center of the strings. The maximum power, on the other hand, is presented below.
