A FOUR-WALL GAME SUCH as racquetball, squash or handball demands of the player a great deal of skill in judging angles and bounces. The ball comes off the wall in a direction determined by the physics of the collision. An understanding of this physics enables a player to predict the ricochet of a ball approaching him and to calculate the ricochet he would like to achieve in order to put the ball out of the reach of his opponent. In discussing these phenomena I shall call to my aid some strange related tricks that can be demonstrated with a highly elastic solid ball sold in toy stores.

The toy ball is almost perfectly elastic: if you drop it, it bounces back nearly all the way to your hand. (A perfectly elastic ball would return to its initial height.) The ball also has a rough surface, so that when I throw it along the floor, it does not slip. Because of the ball's elasticity and roughness, it can be bounced in some surprising ways.

When I throw the ball downward at an angle, it bounces across the floor in a w repeated pattern of high, short hops and low, long hops. If I put some spin on the ball as I throw it, it bounces to the left and right until it runs out of energy. The most startling demonstration involves throwing the ball under a table. A smooth ball would bounce between the table and the floor until it reached the far side of the table. A rough elastic ball bounces back to the thrower.

To study how a ball collides with a surface I first considered a uniformly solid ball bouncing on a floor. Suppose, the ball approaches the floor moving to !t the right and downward. It helps to describe the velocity as being in two parts, one part parallel to the floor and the other part perpendicular. In addition the ball can be spinning about its center. A clockwise spin is a negative rotation and a counterclockwise spin is positive.

The ball's kinetic energy is in three parts, one part for each component of the velocity and one for the spin. If the ball is completely elastic, the collision does not change the total kinetic energy.

(The total kinetic energy is said to be conserved.) Only an ideal ball and collision follow this rule. In practice some kinetic energy is lost by being converted into other forms of energy. For example, some of it might end up in the vibrations of the ball. I shall ignore such losses and concentrate on the movements of a totally elastic ball.

The collision of the ball with the floor changes the perpendicular velocity in a simple way: it reverses the direction but leaves the magnitude and the associated kinetic energy unaltered. The parallel velocity and the spin are altered in more complicated ways. Still, the total kinetic energy is unchanged. An elastic collision might decrease the spin, but the parallel velocity would then be increased just enough to keep the total kinetic energy constant. This requirement of conserving the total kinetic energy is a strong tool for predicting the rebound.

Another important point is that the total angular momentum is conserved. One contribution to the angular momentum comes from the spin. This contribution is equal to the rate of spin multiplied by the ball's moment of inertia. The spin angular momentum is considered to be negative if the spin is clockwise and positive if it is counterclockwise. The moment of inertia depends on the mass of the ball and the way the mass is distributed. For a solid ball of uniform density the moment of inertia is two-fifths of the product of the mass and the square of the radius.

The other part of the angular momentum depends on how fast the ball is moving parallel to the floor at the instant it touches the floor. This contribution to the angular momentum is equal to the product of the ball's mass, the parallel velocity and the radius. If the parallel velocity is toward the right, the contribution is negative; toward the left it is positive. The collision may change the two contributions to angular momentum in both magnitude and sign, but the total angular momentum remains. In sum, regardless of how the ball is thrown to the floor or how it spins, the total kinetic energy and the total angular momentum must remain constant in an ideally elastic collision.

The easiest demonstration is to drop the ball to the floor. If it has no spin initially, it must bounce back to your hand without spin because of the conservation rules. The only kinetic energy it has is associated with its perpendicular velocity. Since that velocity is only reversed by the collision, without any change in magnitude, the kinetic energy is unchanged. None of it can be transferred to the spin or to parallel velocity, and so the ball must travel straight upward. This result also satisfies the requirement that angular momentum be conserved. Before the collision and after it the ball's angular momentum is zero.

Suppose you put a clockwise spin on the ball. The collision directs the-ball onto a new path. At the collision with the floor the spin creates a friction force toward the right, reversing the direction of spin. Because of the friction force the ball also acquires a parallel velocity so that it-bounces to the right. The energy for the parallel velocity is taken from the energy of the initial spin.

Energy is also transferred when the ball is thrown to the floor at an angle and without spin. I had expected the path after such a bounce to be just as steep as the initial path, but it is steeper because the collision red uces the parallel velocity, converting some of its kinetic energy into spin energy: In terms of angular momentum the collision reduces the amount associated with the parallel velocity and increases (from zero) the amount associated with the spin. The total kinetic energy and the total angular momentum are conserved.

The steepness of the path after a collision depends on the initial steepness and the spin. When the initial spin is negative (clockwise), the final steepness is less than it is when the ball is thrown down without spin. A strong spin directs the ball along a low path over the floor. When the initial spin is positive (counterclockwise), the ball may bounce forward in a steep path, upward perpendicular to the floor or even backward, depending on the strength of the initial spin. The bounce is straight up if the ball initially has just the right amount of positive spin. (The product of the spin and the radius of the ball must be equal to three-fourths of the ball's initial parallel velocity.) With more counterclockwise spin the ball rebounds to the left. If the spin is less than the threshold amount,: equal to zero or negative (clockwise), the rebound is to the right:

The steepness of the rebound can be understood in terms of the friction where the ball touches the floor. The friction force is opposite to the direction in which the surface of the ball is moving. At the moment of contact the surface motion has two sources: parallel velocity and spin. The friction opposes the sum of these two motions. For example, if the ball is thrown down at an angle and without spin, the surface touching the floor is moving to the right: The friction force acting on the surface is toward the left, which reduces the parallel velocity. The ball bounces toward the right with less rightward velocity than it had before the collision. Since the amount of perpendicular velocity is unaltered by the friction, the ball bounces in a path steeper than the one it followed in approaching the floor.

I also considered events in which the ball makes several bounces on the floor. Suppose the ball is thrown-to the right without spin. The first bounce reverses the perpendicular velocity (so that the ball goes upward), decreases the parallel velocity and imparts a clockwise spin. The ball rises to its maximum height and falls back to the floor. The-surprising feature is that this second bounce restores the initial spin (which was zero) and parallel velocity. The result is the same regardless of the initial values of spin and parallel velocity. If the ball continues to bounce along the floor, its initial values of spin and parallel velocity are restored after every even number of bounces.

The phenomenon was readily apparent in the action of the elastic toy ball. I painted the equator of the ball so that I could monitor the spin. When I threw the ball to the floor with no initial spin the first bounce was high and short, so

that the ball did not move very far horizontally before the-next bounce. The spin was clockwise. The second bounce was low and long. The ball had essentially no spin. Thereafter the ball repeated the pattern of a high, short bounce followed-by a low, long one. Since the ball was not totally elastic, each bounce was less energetic than the preceding one. A perfectly elastic ball would periodically resume its initial spin of zero and its initial parallel velocity.

The interactions of spin and parallel velocity account for the-strange actions of a ball thrown to the floor so that it strikes the underside of a table.-If the ball is initially without spin, it bounces from the floor on a steep path with a rapid clockwise spin; when it hits the table, it rebounds to the left with a counterclockwise spin. The second bounce from the floor is also to the left with a counterclockwise spin. The perpendicular velocity has been reversed three times but is unchanged in amount. The parallel velocity is now toward the left and is almost unchanged in amount. .Hence the ball almost return6 to the launch site.:

Suppose the ball were smoother and less elastic. The first bounce would result in a weak spin and the second (from the underside of the table) would not be to the left; The ball would continue to travel to the right until it exhausted its kinetic energy.

I next turned my attention to an ideally elastic, hollow racquetball. Such a ball should perform all the tricks of a solid ball, although the spin values differ because the hollow ball has a different moment of inertia. If the ball is thrown at an angle to the floor (toward the right), it will hop straight up provided the spin is counterclockwise and the product of the spin and the ball's radius is equal to one-fourth of the parallel velocity rather than three-fourths.

In racquetball the serve comes off the front wall of the court. The ball rebounds to the opponent either directly or by bouncing from the side walls. The opponent must return the ball to the front wall before it bounces twice an the floor. Except on the serve, the ball can also be bounced from the back wall and the ceiling. I shall consider the shots that are allowed after the serve.

A player can impart spin to the ball with the racquet in only two ways: by stroking forward and over the top of the ball (achieving topspin) or forward and along the bottom of the ball (achieving backspin). Figure 7 [below] depicts the spins from

a view on the right side of the court. Consider a ball hit hard and low toward the front wall with topspin. The collision is similar to one I described for a solid ball. The topspin (clockwise in the illustration) creates an upward friction force that directs the ball upward and reverses the spin. When the ball returns to the floor, the counterclockwise spin forces a low bounce toward the rear of the court. The potential advantage of such a shot is that your opponent may not expect the high rebound from the front wall or the low hop from the floor.

If you hit the ball hard and low toward the front wall with backspin, which is counterclockwise, it bounces toward the floor with a clockwise spin. It hits the floor close to the front wall and rebounds steeply upward. The potential advantage of this shot is that your opponent may not be able to reach the ball before it bounces from the floor a second time.

Usually my stroke gives the ball little or no spin, but it ends up spinning as soon as it bounces from a wall or from the ceiling. Consider a ceiling shot, which I often make to change. the pace of the game. My opponent must adjust not only to the new path but also to strange hops off the floor. Suppose I make the ball bounce from the front

wall to the ceiling. It leaves the ceiling with a clockwise spin. When it hits the floor, its parallel velocity is sharply reduced, making it bounce almost straight up. My opponent, who is expecting a rebound path resembling the path of the approach to the floor, waits too far back in the court.

If I make the ball bounce from the ceiling to the front wall, it approaches the floor with a counterclockwise spin. The collision with the floor increases the parallel velocity, sending the ball into a low hop. Again my opponent misjudges the rebound path and misses the ball. Both ceiling shots are better if I start them from about midcourt. Then the spin as the ball approaches the floor is strong and the strange-hop is enhanced.

Suppose the ball is bounced off the front wall so that it moves toward the left side of the court. If you take an overhead view and ignore the curvature of the path due to gravity, the arrangement is similar to the one in which a solid ball is thrown at an angle to the floor. The collision reverses the perpendicular velocity (in this case the velocity perpendicular to the front wall), decreases the parallel velocity (the velocity toward the left side wall) and imparts a clockwise spin. In the overhead view the final path is steeper with respect to the front wall than the initial path because of the reduction in the parallel velocity. An opponent can quickly learn how to deal with this type of rebound in racquetball.

A more difficult shot to anticipate is one that bounces from two walls. Consider an overhead view of a shot in which the ball bounces from the front wall and then from the left sidewall. The first bounce gives the ball a clockwise spin and a velocity directed toward: the rear wall. Can you make the ball rebound from the side wall in any direction you choose or is the final angle of rebound fixed? Can the final spin be zero or any value of clockwise or counterclockwise rotation? To answer these questions I employed some mathematics published independently by Richard L. Garwin of Columbia University and George L. Strobel of the University of Georgia.

Assume the ball is launched toward the front wall with no spin and has a small initial perpendicular velocity. You can make such a shot if you are near the front of the right side wall. Then an ideally elastic racquetball rebounds from the left side wall at an angle of about 12 degrees. If you are closer: to the center of the court, the initial perpendicular velocity is larger and the angle of the rebound from the left side wall is smaller; the ball travels along the wall to the rear of the court.

You can use this arrangement to advantage. Suppose your opponent is near the middle of the right wall. By bouncing the ball off the front wall and into the left side wall so that it travels along the side wall to the back of the court, you can make it almost impossible for him to return the shot. Even if he is not far from the final path of the ball, the rebound off the side wall might at least prove confusing.

When I tested my calculations with a real racquetball, I found an approximate agreement. The steepest angle of rebound from the side wall was larger than the 12 degrees I had predicted As I increased the initial perpendicular velocity by moving from the right wall toward center court, the angle of rebound decreased until the ball almost hugged the left wall on its way to the rear of the court.

The discrepancy between the actual and the predicted rebound off the side wall arises from the inelastic collisions of a real racquetball. If the ball hits a wall squarely, it compresses uniformly, storing its energy as elastic potential energy. Only part of the energy is reconverted into kinetic energy as the ball pushes off from the wall, again taking the shape of a sphere. A racquetball might bounce back with 60 percent of its energy in such a collision. The perpendicular velocity would then be about 80 percent of the initial value. (The change in velocity is proportional to the square root of the change in energy.)

A glancing collision is more difficult to interpret because the compression of the ball is not uniform and depends on the angle of the collision. The loss of kinetic energy and angular momentum reduces both the spin and the parallel velocity. (When the ball skims along the wall or the floor in an extreme glancing shot, you can hear the energy loss as a high-pitched squeal as the ball skips over the surface.) In my calculations I chose to reduce the spin and the parallel velocity after a collision by .4. With these reductions I found closer agreement between my predictions and the actual rebounds.

I was also able to explain why a real racquetball does not return to me when I throw it under a table. The reductions in energy and angular momentum in the bounces from the floor and the underside of the table trap the ball into bouncing almost vertically until it exhausts its kinetic energy.

Is there a way to hit the ball to the front wall so that it rebounds from a side wall parallel to the front wall? With such a shot you could win every game because your opponent could not possibly get to the ball in-time. As it turns out such a shot is impossible. A rebound from a side wall is always toward the rear of the court.

Can a rebounding ball have any direction of spin or even no spin? Yes, because its final spin depends on the initial ratio of perpendicular and parallel velocities. For a perfectly elastic racquetball a spin of zero results when the ratio is 1 to 5. A smaller ratio yields a clockwise spin (from an overhead- view), a larger ratio a counterclockwise spin.

The Z shot is a three-wall rebound that is marvelous to watch. When it was first introduced in the early 1970's, it confounded even the most experienced players. The ball is hit to the top left side of the front wall, bounces to the left side wall, crosses the court to the rear of the right side wall and then rebounds parallel to the back wall. An opponent will need experience to anticipate the final rebound, but even then the ball will be difficult to return to the front wall. If I hit the Z shot less than perfectly, the ball might still be difficult to return if it hits the floor and then the back wall. My opponent must catch it near the back wall before the ball makes its second bounce on the floor.

Initially I thought a perfect Zshot was impossible. I doubted that the final rebound could be made to move parallel to the back wall. Armed with my mathematics I set out to follow the bounces.

I immediately met a problem. If the ball is assumed to be perfectly elastic, it rebounds from the left- side wall at such a small angle that it hits the back wall instead of the right side wall. I factored an extra-long Court into my calculation. I also ignored the curve resulting from gravity and made the calculation as though the ball remained in a plane parallel to the floor.

To launch the Z shot a player stand near the right wall at about midcourt The ball is hit to the top left side of the front wall about three feet from the corner and three feet from the ceiling. Since such a shot makes the ball leave the left side wall with a clockwise spin, its collision with the right side wall creates friction force toward the front wall.

Consider the velocity and the spin o the ball just before and just after the collision with the right side wall. The perpendicular velocity is reversed, directing the ball toward the opposite side wall. What happens to the spin and the parallel velocity? The collision is similar to one I considered earlier. The friction during the collision opposes both the spin and the parallel velocity, reducing the parallel velocity and reversing the spin. Under the proper conditions the parallel velocity can be reduced to zero, so that the ball's path is perpendicular to the side wall. This is how a perfectly executed Z shot makes the ball travel parallel to the back wall.

When my calculations include the loss of energy with each collision, my predictions are closer to the actual path of a Z shot in a court of the proper dimensions. The possibility of a final rebound parallel to the back wall is still present. My calculations are flawed, however, since the actual path has three dimensions' My assumption of a flat trajectory simplifies the calculations because the axis about which the ball spins is always kept parallel to the wall. In the actual flight of the ball the spin axis is often at an angle with respect to the side walls.

The around-the-walls shot also hi three walls. The ball is bounced from the right side wall to the front wall an then off the left side wall The shot is designed to confuse an opponent, but if the ball ends up at midcourt, he may have an easy chance of returning it to the front wall. I wondered if there was any way I could set up the around-the-walls shot to make the ball rebound from the left side wall parallel to the front wall. Expecting the ball to come to the rear of the court, my opponent would surely be caught off guard by this strange rebound.

I tried the shot in many ways without success. I wondered if the problem was my lack of playing skill, and so I turned again to mathematics. My calculations showed that such a rebound is possible; if the ball begins with much energy and makes a small angle with the right side wall. If I had made the calculations earlier, I could have saved myself many futile swings of the racquet.

Many more shots can be studied with either a solid ball or a hollow racquetball. Perhaps there are some clever shots that even the professional racquetball players have yet to discover. You may be interested in studying how a ball loses energy in a glancing collision with a wall. You may also be interested in following the flight of a ball in three. dimensions, so that the spin axis is no longer parallel to the walls. For this purpose a computer simulation of racquetball would be helpful. Be careful if you experiment with a solid, highly elastic ball in a racquetball court. I tried it just once. The ball moved and rebounded so fast that all I could do was get out of the way.

Bibliography

KINEMATICS OF AN ULTRAELASTIC ROUGH BALL. Richard L. Garwin in American Journal of Physics, Vol. 37, pages 88-92; 1969.

THE DYNAMICS OF SPORTS. David F. Griffing. Mohican Publishing Company, Loudonville, Ohio; 1982.

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