# Ball Spin During Bounce

The bounce of a tennis ball greatly affects the way in which the receiver can return it. Much as the spin plays a large role in the trajectory of a ball, it also strongly affects how the ball bounces. Knowing how spin affects a ball's bounce is a key part of the game of tennis.
In order to analyze the motion of a tennis ball before, during, and after its bounce, its movement must be broken down into separate horizontal and vertical components. In addition, for our purposes we will continue to identify rotational motion of a tennis ball in the direction of topspin as forward rotation/spin, and rotational motion in the direction of underspin as backward rotation/spin. The change in the vertical and horizontal components of a tennis ball's velocity during the bounce are different for each type of shot: flat, topspin, and underspin. It is the ratio of the ball's vertical to horizontal speeds after the bounce that determines the angle at which the ball bounces off the court.
There are two physical properties that are highly important in understanding the dynamics of a tennis ball bounce. The first of these is the coefficient of restitution, or COR, of the ball on the court. The COR is determined by dropping a ball onto the court, and it is the ratio of the relative speed of of the ball just after colliding with the court to its relative speed just before colliding with the court. The COR affects a ball's vertical speed after its bounce. The second important property is the coefficient of kinetic friction, or COF, between the court and the ball. The force of friction of the court on the ball affects the balls spin and horizontal speed, but plays not role in determining its final vertical speed after the bounce. For all tennis courts, the COR and COF of a tennis ball on the court are between 0.5 and 1.0.
Even though a tennis ball is only in contact with the ground for between 5 and 7 milliseconds, the motion dynamics of the ball change dramatically during that time. The explanations given below are not an exhaustive account of the full dynamics of a bouncing tennis ball, but are instead meant as a survey of the underlying bounce properties of tennis balls with varying spins.

Source: [4]

### Spin Change:

When a tennis ball impacts the court, it initially skids and the court exerts a kinetic friction force on the bottom of the ball in the direction opposite that of the ball's horizontal motion (except in a special case of topspin, which is discussed below). This friction force exerts a torque about the center of the ball that causes forward rotational acceleration on the ball. This results in the ball taking on topspin as it leaves the bounce, if it did not already have some. For balls hit with underspin, this results in a complete reversal in direction of the spin of the ball during the bounce. Balls that already have topspin will generally have their topspin increased during the bounce, but it can also be decreased depending on the rate of rotation of the ball. This will be further elaborated on in the Topspin section.

Source: [4]

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### Flat shot:

A tennis shot hit with no spin, or a flat shot, will very nearly reflect off the surface of the court during its bounce. This means that the angle at which the ball impacts the ground, the angle of incident, is approximately equal to angle at which it leaves the ground, the angle of reflection. The ball loses vertical speed because the COR less than 1.0. In addition, the ball loses horizontal speed because the friction force of the court on the ball acts in the direction opposite of the horizontal movement of the tennis ball, and the COF of the ball on the court is greater than 0. This results in the ratio of the vertical speed to the horizontal speed of the ball before and after the bounce being very close in value. Hence, the ball leaves the bounce with approximately the same angle as its angle of incident, but with a slower speed due to smaller horizontal and vertical velocity components. As discussed earlier, the ball have some topspin after the bounce.

Source: [4]

### Topspin:

Img Src: http://www.tennisindustrymag.com/issues/200404/images/200404ball_bounce.png

When a ball hit with topspin impacts the ground, the forward spin of the ball reduces the effect of the friction force of the court on the ball. If the tangential velocity of the top of the ball is less than the horizontal velocity of the ball, then while the ball is in contact with the court during the bounce, the force of friction increases the ball's forward angular velocity. Once the tangential velocity of the top of the ball is the same as the overall horizontal velocity of the ball, the ball begins to roll, which reduces the friction of the court on the ball dramatically. This causes a smaller net friction force to be applied to a ball with topspin during its bounce and results in a ball losing less horizontal speed. The smaller decrease in horizontal speed of the ball causes the ratio of vertical speed to horizontal speed to be lower after the bounce. This results in the ball to leaving the ground at an angle lower than its angle of incident. Further, since both the vertical and horizontal speeds of the ball are decreased during the bounce, the overall speed of the ball is also decreased.
In the case that the tangential velocity of the top of the ball is the greater than the overall horizontal velocity of the ball, the friction force of the court on the ball will instead point in the direction of the overall horizontal velocity of the ball. This will reduce the angular velocity the ball until the tangential velocity of the top of the ball is the same as the overall horizontal velocity of the ball, at which point the ball will begin rolling and the friction force of the court on the ball will be greatly reduced. The force of friction results in an increase in horizontal speed of the ball. Thus the ratio of vertical speed to horizontal speed of the ball is again lower after the bounce. Therefore the ball still leaves the ground at an angle lower than its angle of incident.
Although the angle at which topspin balls leave the bounce is less than their angle of incident, people do not perceive topspin bounces as such. Topspin shots are often used to create a high bounce that is difficult to return. Because balls hit with topspin are usually hit higher over the net than other shots and have a downward Magnus force acting on them, they generally impact the court with a greater vertical speed than other shots, resulting in an overall higher bounce. Thus, we perceive balls hit with topspin to have a higher effective bounce, despite the fact that the spin of the ball reduces the angle at which the ball leaves the ground.

Source: [4]

A Shot Hit With Topspin

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### Underspin:

Img Src: http://www.tennisindustrymag.com/issues/200404/images/200404ball_bounce.png

When a ball with underspin impacts the ground, the ball skids across the surface of the court, and the court exerts a kinetic friction force on the ball in direction opposite that of the overall horizontal motion of the ball. Since the ball has backward rotation, the kinetic friction force must act for a longer period of time on the ball before it will start rotating forward. This increases the net friction force on the ball and reduces its horizontal speed by a greater amount than flat and topspin shots. As a result the ratio of vertical speed to horizontal speed of the ball is higher after the bounce than before it. Thus the ball leaves the ground with an angle greater than its angle of incident. Further, since both the vertical and horizontal speed of the ball are decreased, the overall speed of the ball is also decreased.
Similar to topspin shots, we perceive the bounce of of an underspin shot differently because of how they are used in tennis. Although the angle that an underspin shot leaves the ground with is greater than its angle of incidence, underspin shots are often thought of as having a low bounce. This is because underspin shots are usually hit low above the net and have upward Magnus force acting on them, which causes them to impact the court with a lower vertical speed than other shots, resulting in an overall lower bounce. Thus, we perceive balls hit with underspin to have a lower effective bounce, despite the fact that the spin of the ball increases the angle at which the ball leaves the ground.

Source: [4]

A Shot Hit With Underspin

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