The Rocket Science of Tennis and Its Racket

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Its Not Rocket Science, Its Racket Science Remember warm Sunday afternoons when everyone loaded into the car with the intention of spending some family time together? For me, being one of eight children, it was all too familiar. The entire trip was nothing but fighting over what we were going to get, and who got to pick the cereal. Once we got to the popular cereal isle my brothers, sisters, and I were in constant battle deciding between Cocoa Puffs and Trix. What did we end up with though? Kaboom or something generic like that. It was the same thing (that is what my parents always said), but it never really tasted the same. Imagine my fright when I announced that I was going to join the tennis team and I needed a racket. Just like the cereal, I knew that my parents were interested in saving money. Quality was not in the budget. I had envisioned the Radical Tour 260, the latest and most practical tool for the game of tennis. My parents, on the other hand, had different intentions. We were off to WAL-MART to find the most economical racket that reasonably fit into the budget. At least, that is how my parents explained the situation. Otherwise, in normal language, we were going to pick out the cheapest racket on the shelf. I began arguing, stressing the importance of how the racket affected my skill on the court. I continued rambling and whining, and with that my father issued a challenge: If I could find scientific research backing up my reasoning for needing the Radical Tour 260, he would be sold. My need for that racket was overwhelming. I did not want to be the only guy on the team without the racket. It just wouldn't be fair. With that thought, I ran off to the library to start researching. This, my report, is what I gave my parents the next evening. To determine how important the racket is in the success of a tennis player, one must first understand the basic motions of the ball, the many swings affecting the ball, the anatomy of the racket, and how, through the laws of physics, the racket and its actions can be manipulated to ensure success in even the beginning tennis player. To achieve a full understanding of how physics affects the game of tennis, I will begin with defining a few basic physical principles that influence motions of the ball. Next, I will apply these definitions to several physical characteristics such as the coefficient of friction, speed, resistance, Newton's Laws, Magnus force, gravitational pull, and the conservation of momentum. Finally, I will use these characteristics to describe how and why the technology of tennis rackets has changed in recent years. The motion of a tennis ball through air is determined by the laws of physics. The way in which the ball goes over the net on a serve is not as simplistic as it might sound. It includes velocity (both final and initial), acceleration of the ball, forces acting on the ball and the angles of motion during the swing and the follow through. Speed is a ratio between the displacement divided by the time it took for the displacement to occur (v=d/t). For example, imagine a tennis player hits a ball ten yards in two seconds. The average speed of the ball is five yards per second. At some point, the ball may have been going faster or slower than five yards per second, but again it is the average speed. When the velocity of the ball changes, the ball undergoes acceleration. Acceleration is the change in velocity divided by the interval of time. When the tennis ball's velocity and acceleration are in the same direction, the speed of the ball occurs with time. When the ball's velocity and acceleration are in opposite directions, however, the speed of the ball decreases with time. Once the ball is first shot into the air, the laws of physics take over and determine where it will go. There is nothing that the player or his or her opponent can do to guide it or change its path. There are three forces acting on the ball during its flight; gravity, air resistance, and the Magnus force which causes the ball to curve. The force due to gravity (mg) is always pointed straight down toward the Earth. Air resistance slows the ball, and in the range of speeds encountered in tennis, the force it causes is proportional to the square of the ball's speed. For example, a ball moving at 50 m.p.h. will encounter four times as much air resistance force than that of a ball moving at 20 m.p.h. Wind also creates an air resistance force, which can be analyzed in a similar manner. Because air resistance force is proportional to the square of the speed, a crosswind of 20 m.p.h. will exert four times as much force on the ball as a 10 m.p.h. crosswind, and a 30 m.p.h. crosswind provides a force nine times as strong as the 10 m.p.h. wind. This is obvious when a tennis player tosses the ball up for a serve if there is a brisk breeze. The Magnus force is at right angles to the direction that the ball is moving and is proportional to how fast the ball is spinning. It is also proportional to the square of the ball's speed. Because of these factors, it is very important for tennis players to be able to observe these certain characteristics. They must be able to think critically to place the shot in the correct side of the opponent's court. There are many ways in which a player may hit the tennis ball. Choosing a good strategy and position, hitting high-percentage shots, and using the proper equipment may help the player win more points. The angle of the racket face and the direction of the racket velocity at the instant of contact between the ball and the racket determine where exactly the ball will go. When a player stands at the forehand corner of the court and attempts to return a shot to the center of the challenger's court with a forehand drive, the shot will go crosscourt if the player swings a little early. If he or she swings a little late, the shot will go down the line (Cantin 6). The swing of a tennis racket can be described as the arc of a circle. At the second that the player hits the ball, the racket is in a certain position in the arc. Thus, the face of the racket is pointing in a certain direction, and at that moment the racket is moving tangent to the arc. The angular error of the racket is given by the formula 57 x timing error x (ball speed + racket speed)/ swing radius. This means that the worse the timing error, the larger the angular error. This error decreases as the swing radius increases, but it increases as the racket speed and the speed of the approaching ball increase. This attributes to the knots in a tennis player's stomach as the opponent puts increased pressure on them. Increasing the radius of swing however, will improve the player's accuracy and control. If the player keeps a firm wrist and uses his or her shoulders as the pivot point for his or her shots, he or she will double the radius of his or her swing and will reduce by half the horizontal angular error caused by the timing error associated with that shot (Brody 119). The three most popular techniques in the sport of tennis include topspin, backspin, and sidespin. Topspin is, by far, the most challenging and requires a greater appreciation of physics. Topspin on a tennis ball is usually called the powerspin. The difference between a shot with topspin and a shot without topspin is rotational motion on the shot with topspin as well as translational motion. If the face of the racket is oriented so that it is perpendicular to the direction of the racket's motion, the resulting shot will have little or no spin. So how do you generate a lift and spin on the tennis ball? Lift is generated by creating a pressure difference and deflecting the flow. To create a pressure difference on the ball, it needs to move more fluid around one side than the other. Spinning the ball will set up the imbalance, thus making the pressure difference. When the tennis ball rotates, the fluid that is in contact with the ball's surface tends to rotate with the ball. The air next to the air on the surface tends to do the same thing. Far from the ball, this rotation does not affect the surrounding air. Very close to the ball, however, these fluid layers make up what is called a boundary layer. Consider the topspin stroke; if the ball doesn't rotate as it flies through the air, then both the top and bottom sides of the ball meet the air rushing over it at the same speed. Relative to the ball, the top of the ball in topspin spins forward into the oncoming air. There is more movement of air towards the bottom surface. Now, more fluid needs to pass through the same space on the underside of the ball. Basically, the flow is squashed on the lower side of the ball. This means that there needs to be a higher velocity on the lower side of the ball, and, subsequently, a lower velocity on the top of the ball. On the top side of the ball this lower velocity creates a higher pressure. This effect is known as Bernoulli's Law. With high pressure on one side and low pressure on the other, there is an imbalance in the forces on the ball. In the case of topspin, the higher pressure on the top curves the ball downward from its straight line path. Finally, to execute full understanding of topspins, one must be able to identify rotational momentum and how it differs from other shots in tennis. Rotational motion is the spinning of the ball as it sails across the net. Pure rotational motion describes the principle that all points in the ball move in circles, and that the centers of these circles all lie on a line called the axis of rotation. Because each point rotating with the ball has a different linear velocity, spinning causes more air to flow over the top of the ball and thus the ball falls shorter. If an object has points on it spinning, it has an access of rotation which is located in the center of the ball. Backspin and sidespin are also two other techniques in tennis, however, they are not as interesting or as challenging as the topspin. Backspin is accomplished by chopping at the ball with an upward tilt of the racket. The ball will be moving up, and will remain high. The backspin shot floats the longest, and bounces very close to the baseline. Thus, by successfully executing a backspin, a player reduces the margin for allowable error (Bloom 2). Sidespin is yet another popular technique in the game of tennis. Sidespin on a tennis ball makes the ball appear to be moving to the left or right. Not only will the tennis ball look like it's moving to the right or left, but it will remain low when crossing the net. Spin is applied to the ball by the friction between the ball and the strings when the ball slides or rolls across the racket face. The distance that the ball slides or rolls across the racket is determined by the dwell time and the velocity of the racket in the direction parallel to the racket face (Randall). The height to which the ball bounces and the speed of the court are also subject to those same laws. Tennis courts are made of all types of surfaces: clay, grass, concrete, asphalt, and rubber. When a ball bounces on the court, its horizontal speed is reduced by its interaction with the court's surface. If the ball slows down a great deal upon bouncing, the court is slow, while a fast court does not affect the ball's horizontal speed as much. There are two characteristics of a court surface that influence the ball as it bounces. These characteristics are the coefficient of restitution and the coefficient of friction between the ball and the surface. The coefficient of restitution determines how high the ball will bounce from a certain height. It is defined as the "ratio of vertical ball speed after the bounce to the vertical ball speed before the bounce" (Brody 62). A high coefficient of friction is a measure of the frictional force of the sort of surface on the tennis ball in a direction parallel to the surface; it usually slows a ball down (See figure 1). A high value of the coefficient of friction means that the frictional force on the ball is large. While coefficient of restitution influences the vertical velocity of the ball, the friction affects the horizontal velocity of the ball, and that is the direction that determines a court's speed (Brody 63). The larger the friction between the ball, the more the ball will slow down when it bounces, and the slower the court will be. When a ball with no spin hits a court surface, there is a frictional force parallel to the surface and in a direction opposite to the ball's direction of motion. The ball will begin to slide or skid along the court, with the bottom of the ball slowing down more than the rest of the ball; this will cause the ball to rotate. If the frictional force is powerful enough and the ball's incident angle of bounce is large enough, the ball will begin to roll on the court surface before it rebounds and loses contact with the ground. If the ball leaves the court before rolling begins, it is considered to be a fast court. Aging of the court also determines the speed of a court. Many hard courts must be resurfaced if the slowness that they have when they are new is to be retained. These courts are covered with a latex that contains sand. The roughness of the sand creates a great deal of friction between the surface and the ball. As the court is played on, however, constant wear tends to smooth the surface, reducing the friction. As a result, the court speeds up with age and use. After gaining an understanding for the motion of the ball and the many forces it encounters while in the air and on the court, it is important to understand the general "anatomy" of a tennis racket and how to use its features to fully benefit a one's game. Most of tennis racket science is involved with technological improvements of the rackets in order to improve performance on the court, much like my Radical Tour 260. Changes in the racket have included composition of frames, string pattering, vibration-dampening systems, and the overall head size. Wooden rackets were originally used until the early 1980s when it was discovered that graphite produced stiffer rackets, thus increasing the power. Moreover, the enlargement of the head has been the most beneficial in terms of performance. The basis of increasing the head size was to enlarge the sweet spot, the precise area on the racket face that delivers the most powerful shot with the least amount of vibration. Experiments by racket maker Howard Head, the developer of the idea of larger heads for graphite rackets, revealed that "increasing the face size by twenty percent increased the sweet spot by nearly three hundred percent" (Brody 213). A very practical question to ask a tennis player is what is the ideal racket? This is the same question I asked myself and my teammates as I decided that the Radical Tour 260 was the racket for me. One must be aware of the principles of physics that go into designing a high performance racket. These principles include the characteristics of strings, center of percussion, racket vibrations, and moments of inertia. The strings of a tennis racket play an important role in how the ball is hit. There is more to strings than just tension. Years ago, when rackets were strung, the head sizes were all the same and thus, the tension was also. Now, with a various head-sizes, a tension of 65 pounds in a standard racket plays tightly, while 65 pounds in an oversize frame may play too loosely. The way the racket plays with respect to the stings can determine how much of the string plane deforms when a force is applied to the racket. Rackets will play in a similar manner if they are strung so that their curves of string plane deformation versus force are similar. By measuring the string plane deformation, I can compare the Radical Tour 260 with a Wilson Kramer strung with 16- gauge string and know how the strings in one will play in relation to the other (Brody 6). Also, if one increases the tension of the strings in proportion to changes in the length of the strings in the head, the string plane deformation is similar to the first. Simplistically stated, in order to change from one frame size to another while retaining similar playing characteristics from the strings, the tension divided by string length must be kept the same. This is why the oversize racket is strung at higher tensions. One of the many reasons that tennis uses rackets instead of paddles is so that the player can get power. The goal is for the ball to leave the strings with a high velocity without having to swing the racket. The tighter the racket is strung, the more it feels like a wooden board and the less power the player will get. Why do loose strings give more power than tighter strings? Tennis balls do not store and return energy efficiently. For example, imagine throwing a tennis ball from a height of 100 inches onto a hard floor. The tennis ball only rebounds to a height of about 55 inches, a loss of about 45 percent of the initial energy of the ball. Strings, however, are designed to return 92.5 percent of the energy that is fed to them (Watts 84). To give the ball the maximum energy, the strings must store the energy by deflecting. If the strings have a lower tension, they will deflect more and the ball will deform less. So why not string all rackets loosely? By reducing the tension too much, the speed of the ball will be inadequate and the strings will wear out too fast from excessive rubbing. Moreover, by stringing a racket loosely, control must be sacrificed. Reasons for loss of control because of loose stringing includes: making the speed of the ball more dependent upon the pace of the opponent's shot, changing the angle at which the ball leaves the racket, and increasing the dwell time of the ball on the strings. This allows the racket to twist or turn more while the ball is still in contact. The looser the strings, the longer the ball will reside on the strings. The dwell time of the ball on the strings should increase as the inverse of the square root of the tension. In addition, the dwell time of the ball on the strings decreases the harder the ball is hit, because the strings become effectively stiffer the more they are forced to deform (Brody 12). When a player hits a shot and feels great, he or she has hit the sweet spot. According to the American Journal of Physics, there are three sweet spots of a racket (Bloom 4). Sweet spot number one is the initial shock to a players hand. To some this is known as finding the node of the first harmonic (See figure 3). Sweet spot number two is

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