# Ball Spin In Flight

When a ball is hit in tennis, spin is often imparted on it to affect its trajectory and bounce. The three most common ways to hit a tennis ball are flat (no spin), with topspin, or sliced (hit with underspin/backspin). The type of spin put on the ball affects its trajectory in the air, as well as how it bounces. We will will cover the effects and physics of these three types of spin. For our explanations, we will identify rotational motion of a tennis ball in the direction of topspin a forward rotation/spin, and rotational motion in the direction of underspin as backward rotation/spin. First, however, we must cover the Magnus effect, which is what causes the differences in movement behavior during flight of the balls undergoing different spins.

### The Magnus Effect is a physical phenomenon that imparts an additional force on a spinning object. The spinning of a ball causes the drag forces at the top and bottom of the ball to be unequal. In the case of backspin (pictured), the drag at the bottom of the ball is greater than at the top of the ball because the tangential velocity of the bottom of the ball is in the same direction as the velocity of the ball's trajectory. This means the bottom of the ball has a greater magnitude velocity relative to air around it than the rest of the ball, which results an increased drag force. it Similarly, the drag at the top of the ball is reduced since the top of the ball has a lower velocity relative to the air around it than the rest of the ball. These drag forces can be thought of as exerting a unequal pressure on the ball, with greater drag corresponding to greater pressure. The resulting pressure differential then causes a net force on the ball in the direction of the low pressure. In the case of backspin this results in an upward force on the ball, and for topspin this results in a downward force on the ball. This is similar to how an airfoil generates lift.

Img Src: http://en.wikipedia.org/wiki/Magnus_effect

Source: 

### Flat Hit:

A ball hit with no spin, also known as a flat hit, does not rotate with any significant angular velocity as it flies through the air. Thus it does not experience a Magnus force. The ball does however experience an ordinary drag force that acts in the direction opposite of its velocity. As noted on the previous page, the magnitude of the drag force on an object is given by $F_d=\frac\left\{1\right\}\left\{2\right\}\rho v^2 C A.$. The trajectory of a flat shot can thus be approximated numerically using the equations for the horizontal and vertical accelerations of a small projectile experiencing drag:
${a}_{x}=-\frac{\rho CA}{2m}{v}_{x}\sqrt{{v}_{x}^{2}+{v}_{y}^{}}$$a_y=-g-\frac\left\{\rhoCA\right\}\left\{2m\right\}v_y\sqrt\left\{v_x^2+v_y^2$where is the drag coefficient, $\rho$ is the density of air, $A$ is cross-section area of the object, $m$ is the mass of the object, $g$ is the acceleration of gravity, and $v_x$ and $v_y$ are the horizontal and vertical velocities of the project respectively. Unlike projectile motion in a vacuum, the horizontal and vertical components of acceleration are not independent of each other since $a_x$ depends on $v_y$ and vice-versa.

Source: ### Topspin: Img Src: http://twu.tennis-warehouse.com/learning_center/aerodynamics2.php

To hit a ball with topspin, a player rotates the head of the racquet over the top of the ball during the hit. This exerts a torque about the center of the ball that causes the ball to rotate with a forward spin during its flight. As the diagram above shows, topspin is characterized by the tangential velocity of the top of the ball pointing in the same direction as the trajectory of the ball.
Hitting a ball with topspin causes a Magnus force to act on the ball perpendicular to the velocity of the ball in the downward direction. Because there is an additional downward force on the ball, tennis players can hit the ball with a greater speed and have it land in the court, if they apply topspin to the ball when hitting it. In addition, balls can be hit higher above the net, making the shot more difficult to return, because the addition downward force pulls the ball downwards. This also results in a higher bounce for the ball.

Source: Img Src: http://twu.tennis-warehouse.com/learning_center/aerodynamics2.php

### Underspin (Backspin):

To hit a ball with underspin, or backspin, a player angles his racquet back and slides it underneath the ball when hitting it. This type of shot is called a slice. It exerts a torque about the center of the ball that causes the ball to rotate with a backward spin during its flight. As the diagram above shows, underspin is characterized by the tangential velocity of the top of the ball pointing in the opposite direction as the trajectory of the ball.
Hitting a ball with topspin causes a Magnus force to act on the ball perpendicular to the velocity of the ball in the upward direction. Because there is an additional upward force on the ball, the ball seems to "float" through the air as it flies. Slice shots are thus generally hit low over then net and with a relatively slow speed, in order to keep them from going out of the bounds of the court. This results in the lower bounce for the ball. In addition, slicing the ball allows the player to hit the ball to a precise location in the opponents court. However, due to the way slice shots are hit, if a player attempts to slice a ball that is moving to fast, or hit a slice shot too hard, he or she can easily hit the ball out of bounds.

Source: Img Src: http://twu.tennis-warehouse.com/learning_center/aerodynamics2.php 