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# The Tennis Serve Analysis

This picture was taken at the Gonzaga tennis courts. The picture depicts a tennis player in the motion of serving the ball. The purpose of the serve is to get the ball from the baseline, where the server stands, to the crosscourt service box. The goal of the server is to strike the ball in a way that makes it difficult for the opposing player to return it after one bounce in the opposite service box. The difficulty of the serve return can be increased through a combination of spin put on the ball and manipulation of ball velocity through various techniques.

There are many physics concepts that can be utilized in the explanation of the process of serving the tennis ball. First, the server creates elastic potential energy in the bending of their body. This elastic potential energy along with the ground reaction force due to the pushing of the player’s legs heavily contributes to the total ball speed of the serve. The extension of the radius through racquet extension more significantly maximizes linear velocity. The orientation of the racquet during contact influences the ball path through angle of contact.

The rotation of the racquet about the wrist creates topspin through different magnitudes of force exerted upon the top and bottom portions of the tennis ball. Topspin is the forward rotation of the tennis ball in the direction of ball velocity over the net. Lastly, as the ball travels through the air the Bernoulli principle can be used to explain the air pressure gradient between the air above and below the ball. This gradient generates a force that pushes the ball down at a speed greater than just due to gravity.

As the tennis server prepares to swing their racket in the serve, they bend their knees, which stores elastic potential energy in their legs. This elastic potential energy is also referred to as strain energy. As the legs are bent, the muscles and tendons are stretched in a way that stores this strain energy (Buckerfield, n. d. ). The greater the bend and stretching in the legs, the greater the power generated as the legs are extended (Downward, 2015). As the tennis player extends their legs, this elastic potential energy is converted to kinetic energy and a ground reaction force is generated.

This ground reaction force can also be called normal force. As the player extends their legs, the tennis court ground in contact with their feet is compressed (Colligan, 2015). This demonstrates Newton’s third law which is “for every action, there is an equal and opposite reaction” (Romanov, 2015). This ground reaction force pushes back on the player’s feet and this force is then transferred through the various elements of the body that contribute a portion to the total energy. The legs, hips, trunk, and back in the kinetic chain of the body contribute 54% of the total ball speed of a tennis serve (Kibler, 1995).

This statistic demonstrates how important the base of a tennis player is during their serve. The serve is about more than just arm movement. The ball is tossed and when it reaches its maximum height the server has rotated and extended their arm to maximize the linear velocity at impact. The increase of the radius of the racquet from the body’s axis of rotation slightly lowers angular velocity because it raises the moment of inertia. This is due to conservation of angular momentum, which dictates that if the moment of inertia increases, the angular velocity must decrease (Cutnell et al. 2015).

However, this decreasing effect is small in magnitude in comparison to the increase in linear velocity that is due to the increase in radius of the racquet from the body’s axis of rotation. The product of angular velocity and radius is the linear velocity. Thus, the increase in radius during the tennis serve maximizes the linear velocity of the racquet, even though the angular velocity decreases slightly. Overall, the effect of increasing the radius has a more significant increase in linear velocity than the decrease in angular velocity.

The orientation and rotation of the racquet are significant in the generation of topspin and achievement of maximal tennis ball velocity. First, the tennis racquet tilt due to rotation of the racquet upon contact with the ball must be considered. When executing a topspin serve the server makes contact with the ball as close to the top of the ball toss as possible. If the tennis ball contacts with the strings 0 degrees from the normal of the racquet surface, then the ball will bounce off in the exact same angle the contact was made with.

Although not exactly the same, I have thought of the racquet as a plane mirror and the ball as a ray of light. Thus, the angle the ball makes contact with the racquet can be thought of as the angle of incidence. Thus, if the racquet is tilted slightly forward due to rotation of the swing, then the angle the ball hits the racquet is no longer 0 degrees from the normal. The angle the ball comes off of the racquet after contact thus is at an angle towards the ground, resulting in ball topspin generation (Cross, 2011).

Next, as the racquet makes contact with the ball the racquet rotates approximately 10 degrees about the axis of rotation centered at the wrist. Due to this racquet rotation, the top of the racquet has a higher angular velocity than the bottom of the racquet (Cross, 2011). When the top part of the ball makes contact with the higher point on the racquet (higher angular velocity) there is increased compression of the ball and strings as compared to the lower part of the ball that makes contact with the lower point on the racquet (lower angular velocity).

Figure 1. Force experienced by the ball due to the racquet versus the duration of the contact (Knoll, n. d. ) The figure above displays the maximum force on the ball throughout the contact with the racquet. This force is at maximum halfway through the contact duration when the ball and racquet is compressed to a maximum (Knoll, n. d. ). Thus, the increased compression of the top part of the ball results in that portion of the ball experiencing more force from the racquet than that which the lower part of the ball (less compressed).

To conclude, with the top part of the ball experiencing more force than the lower part, topspin is achieved (Cross, 2011). As the ball travels through the air, the topspin makes the ball dive at a rate faster than just attributed to the acceleration due to gravity. To understand this effect the Bernoulli principle is applied. Bernoulli’s principle explains the relationship between fluid speed and fluid pressure. Fluids can mean liquids and gasses so this principle can be applied to the air surrounding a tennis ball as it travels across the court (Physics Force).

As shown in my drawn figure, from a side perspective the tennis ball travels from left to right with a clockwise rotation (topspin). The air below the ball moves in the same direction as the ball rotation so its speed is higher. The air above the ball moves in the opposite direction as the ball rotation so its speed is lower. According to Bernoulli’s principle the area around the faster moving air below the ball has lower pressure. In the same way, the area around the slower moving air above the ball has higher pressure (Physics Force).

The ball experiences a force in the direction of high pressure region to low pressure region, which pushes the ball downward. This is why the ball displays a “heavy ball” effect as it crosses over the net. The history of tennis is deep and spans across the globe. The first tennis was thought to have occurred in France in the 12th century, but was played without a racquet (The Evolution). In terms of the physics discussed in this paper, this type of play with no racquet would change almost all of the applications. Topspin did not become prominent until the 1970s.

Before then, topspin was not well understood or implemented and thus was thought of as just an interesting spectacle. However, in the 1970s, there arose two players who seemed to properly utilize topspin in terms of the physics based advantage it provided. These two players were Bjorn Borg and Ivan Lendl. They both showed that with topspin the ball could be hit at a greater velocity and still land in fair play (The Evolution). This created the basis for the extreme power the players display in the tennis we see today.

The physics of tennis has been greatly explored by many because of the competitive advantage it can give a player if properly utilized. Through watching many tennis matches throughout my life I have developed an appreciation for the physics concepts that I learned through the research for this assignment. One of my favorite players is Roger Federer and through watching tape of him I have come to the conclusion that his tennis technique perfectly implements many aspects of physics.

For example, Roger Federer is one of the world’s best tennis players and he utilizes the physics of the topspin serve to the fullest. Roger Federer generally aims his serve to the opposite corner of the opposing service box. The speed of his serve is 120 mph (54m/s). With a service speed this fast I wondered what the reaction time of the opposing player would have to be to return the serve. First, I determined the vector displacement of the tennis serve not taking into account his height. Assuming he stands at the halfway point of the length of the baseline his x-displacement would be 4. 5 meters.

Assuming the ball lands in the opposing service box corner the y-displacement would be 18 meters. Using Pythagorean theorem and solving for the hypotenuse the displacement was found to be 18. 45 meters. Taking into account Roger Federer’s height including extended racket of 2. 65 meters, the actual displacement of the ball would be 18. 64 meters. Finally, using the equation Velocity= displacement/change in time, the reaction time was determined to be 0. 35 seconds. Using physics to derive the reaction time sheds light on why Roger Federer’s serve is so difficult to return.

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