Physics 111: Fundamental Physics I: The Physics of Tennis

As an avid tennis player, I can’t help but see all the miraculous physics that manifests on the court. In my childhood I have had phrases like “maintain your racquet-head speed” or “rotate your shoulders” drilled into my head. As I look back on all the instructions and advice given by coaches I have interacted with over the past decade, I can now appreciate the objective rationale that underlie them. Below, I will list some of the most repeated phrases given by my coaches and use physics to explain why they aren’t nonsense.

Advice #1: “Follow through on your shots”

At the contact point between your strings and the ball, you are essentially applying a force over a period of time. This immediately makes me think of impulse. By rearranging Newton’s second law, one can obtain this equation: FdeltaT = deltaP. What this means is that either increasing the time over which the force is applied, or the magnitude of the force itself will increase the ball’s momentum. Any increase in the ball’s momentum will have to imply an increase in the ball’s velocity since the mass of the ball is constant before and after the collision. “Following through” essentially means increasing the duration upon which the ball is in contact the strings. With this in mind, the reason why one must “follow through” on their shots is because it increases the time over which the force is applied, leading to a greater change in the ball’s velocity. The reason why people don’t usually suggest increasing the force of the swing as a means of generating higher momentum is because using too much force can cause you to lose control of the ball, thus increasing the time is more effective.

Advice #2: “Lowering your string tension will increase your power”

When a tennis ball makes contact with the strings, the collision is not perfectly elastic or perfectly inelastic. In order to understand why lowering the tension will increase power, we must analyze the extent to which each object in the collision (the racquet and ball) loses energy in the collision. The coefficient of restitution (COR) describes the extent of kinetic energy lose, and it is defined as the ratio of the relative velocity after the collision to the relative velocity before the collision between two objects. It turns out that a tennis ball has a very low COR, meaning that most of its kinetic is lost during the collision. Figure 1 shows that at ball velocities that are characteristic of intermediate to advanced players ( 30 to 35 m/s), the COR is around .5 which means that balls return only 50% of their kinetic energy. Most of this energy loss is due to the ball deforming upon contact. Strings, however, have a COR of around 90%. Which means that strings return around 90% of their kinetic energy. With this in mind, you would want more impact energy to be stored in the strings as opposed to the ball since the strings return more energy (i.e. lose less energy) upon impact. Thus, to increase your power (i.e. speed of ball post-collision) one must find a way to both increase the energy stored in the springs and decrease the energy stored in the ball (since most energy stored in the ball dissipates). It turns out that lowering the tension of the string accomplishes both of these things at once: a lower tension will cause less ball deformation (which lowers the amount of energy stored in the ball) and increases the amount of energy stored in the strings. This maximizes the amount of kinetic energy returned to the ball post-impact and will increase your power.

Figure 1.

Advice #3: “Don’t keep your racquet too far away from the center of your body”

As any good tennis player knows, rotating your shoulders and uncoiling as you make contact with the ball will allows you to impart more force on the ball. One important concept in rotational motion is your moment of inertia. This generally describes the resistance to rotational motion, meaning that a higher moment of inertia will make it more difficult to change your angular velocity. Anything that brings more of your mass closer to the center of your body will make it easier to rotate into your shot. If we simply the tennis racquet as a point mass and consider the plane that bisects the left and right sides of a body as the axis of rotation, then the moment of inertia of the racquet (under these assumptions) would be defined as MR², where M is the mass of the racquet and R is the distance from the center of the player. By bringing the racquet closer to the center of the body, you are decreasing R. This, in turn, will decrease the racquet’s moment of inertia, and thus the total moment of inertia of the system. Thus, bringing the racquet closer to the body will decrease the system’s moment of inertia, and make the uncoiling phase of the shot easier.

Advice #4: “Increase the spin on your shot to ensure the ball does not go out”

To explain this advice, let’s first explore the mechanism by which spin is imparted on the ball. A tennis racquet is composed two sets of strings – one set runs longitudinally from the top of the racquet to the bottom (“mains”) and the other runs laterally (cross). As a tennis ball makes contact with the strings, the “mains” are deformed downwards (figure 2). The reason why the main strings deform is because modern strings have a very low static coefficient of friction. The deformed string can be thought of us a spring that is absorbing the rotational and translational kinetic energy of the tennis ball into spring potential energy. This spring potential energy is then released as the string snaps back to the equilibrium position (figure 2). This release of spring potential energy is transferred to the rotational kinetic energy of the ball. This transfer of energy is mediated by the torque exerted by the strings. Since the strings exert an upward force on the ball that is perpendicular to the ball’s radial component, the torque would equal the force of the strings multiplied by the ball’s radius. This torque is what makes the ball rotate and have “topspin”. Once the ball is spinning in the air, the ball follows a more downward trajectory than would be predicted by the force of gravity alone. This happens because of the way in which a spinning object interacts with the air. A simple explanation of this phenomenon involves Newton’s third law. A spinning object will exert a force on the air causing an equal and opposite force to be exerted on the object. This is called the magnus effect. For a typical topspin tennis shot, the counterclockwise rotation of the ball will exert an upward force on the air, causing the air to exert an equal and opposite force (a magnus force) downward on the ball (figure 3). This downward force enhances the downward component of the ball’s trajectory which is why increasing the spin on your shot helps the ball stay in.