Bouncing Balls Rod Cross Physics Department Sydney University June 2006 Ball sports
Most major sporting activities involve the use of a ball. Think of football, soccer, baseball, softball, golf, tennis, table tennis, squash, basketball, cricket, billiards, netball, handball, volleyball, hockey and bowling. Only a few sports get by without a ball, such as athletics, swimming, surfing, archery, yachting, skiing and a few others.
The behaviour of the ball in ball sports is learnt by players as a result of hundreds if not thousands of hours of practice and play. How or why it happens is not learnt along the way. The how and why is the subject of this presentation. We will consider:
1. Bounce in the vertical direction
2. Contact time
3. Bounce in the horizontal direction
4. Forces on the ball
5. Ball spin
6. Suggested experiments
Some balls bounce higher than others. In most ball sports the bounce height is specified by the rules of the game for a given drop height onto a given surface. In tennis for example, an approved ball dropped from a height of 100 in (2.54 m) onto a slab of concrete must bounce to a height between 53 and 58 inches when tested at a temperature of 21 C. If it bounces higher or lower than that, players will probably complain that there is something wrong with the ball.
100 inches is just out of reach of most people. If you want to test a ball yourself, there are two perfectly acceptable alternatives (although not officially approved). One is to drop the ball from a smaller height, say 80 or 90 inches. Another is to throw the ball up to about 100 inches and let it fall from whatever height it reaches. In either case, the best way of measuring the drop and bounce heights is to film the bounce with a video camera and then measure the heights from the film. If D is the drop height and B is the bounce height then the ratio B/D must be between 53/100 = 0.53 and 58/100 = 0.58 for a tennis ball.
In practice,B/D for any given ball varies only slightly with the drop height D. If B/D = 0.55 say for a ball dropped from 100 in, then B/D will be about 0.56 when dropped from a height of 80 in, or 0.54 when dropped from a height of 120 in.
The coefficient of restitution (COR) is defined for tests like these as (rebound speed)/(incident speed). Since fall and bounce heights are proportional to the speed squared, the formula is
COR = Square Root of (B/D)
If B/D = 0.56 then COR = 0.75. The COR drops slightly as the incident speed increases
1. Measure the drop and bounce heights of a tennis ball using a long ruler or graduated stick. You can measure the heights from the top or the bottom of the ball. It doesn’t matter which, provided you are consistent. What problems do you encounter? eg hard to pick the actual bounce height, ball is not round and doesn’t always bounce vertically or on the same spot etc.
2. Repeat using a digital video camera to film the ball and to transfer clips to a computer.
3. Vary the drop height D and plot a graph of B/D vs D. This will tell you whether the drop height is critical when testing a ball or whether it doesn’t matter very much. If it does matter then the graph will allow you to convert your own preferred drop height into “official” results.
4. Bounce off different court surfaces (soft and hard) and compare. How do different courts compare? A ball will bounce higher off a hard surface than off grass or carpet.
5. Bounce off the strings of a racquet when the racquet itself is firmly clamped to a solid surface (eg put your foot on the handle so the racquet itself doesn’t bounce around). The ball bounces much higher off the strings than off a solid surface. This is called the “trampoline” effect.
6. Repeat at different temperatures (eg early morning, hottest part of the day). Almost all balls bounce higher when they are hot. By how much? Will it affect your game? On a hot day, an approved tennis ball can bounce to a height of 60 in or more when dropped from a height of 100 in.
When a ball bounces it spends less than 1/100 th of a second in contact with the surface. During the bounce, it squashes and comes to a complete stop. As it springs back to its original shape it pushes itself back up off the surface and jumps off the surface like a person doing a standing jump. It can’t bounce back to its original drop height since energy is lost in the ball when it squashes. Most of the energy loss is due to friction inside the ball and it goes into heating the ball. The ball temperature increases slightly every bounce.
If you film enough bounces with a video camera you might get lucky and catch the ball during one of those bounces while it is squashed. You won’t get enough images of the ball to determine the contact time unless the camera can record at 1000 frames/sec or more. Most video cameras record only 25 or 30 frames/sec. The best way to measure the contact time is to drop the ball onto a thin piezo disk taped onto a solid surface such as a table. Piezo disks about 20 mm diameter and about 0.3 mm thick are used in piezo buzzers and in musical greeting cards. You can extract the disk carefully with a pair of pliers to cut away the plastic housing.
The piezo disk will have two connecting leads which can be connected to a digital storage oscilloscope to measure the voltage generated by the disk. A tennis ball dropped onto the disk will generate a voltage of about 0.5 V for a time of 0.005 seconds. That is the contact time.
Contact time increases with ball mass and decreases with ball stiffness. The contact time of a small steel ball bearing bouncing on steel is about 20 microseconds. The contact time of a basketball or a soccer ball on a wood floor is about 15 milliseconds (0.015 sec). Contact time decreases slightly as the incident ball speed increases since ball stiffness increases the more it is compressed.
In practice, a ball is rarely incident vertically on a surface. It is usually incident at some other angle, such as when a player serves a ball in tennis or returns the ball over the net. In these situations the bounce angle is roughly the same as the angle of incidence but it depends on how much the ball slows down when it bounces. The slowing down effect is due to friction between the ball and the surface. On a smooth, hard surface the ball will slow down by around 20%or 30% in the horizontal direction. On a rough surface, or on a soft surface where the ball digs slightly into the surface, the ball can slow down by around 50% or 60%. In some cases the ball slows down so much that it bounces backward. A tennis ball incident almost vertically with backspin can bounce back over the net. Oval shaped footballs can also bounce backward, especially if they are pointing backward when they land on the ground.
A ball incident vertically on a surface without spin will bounce without spin. A ball incident at some other angle without spin will bounce with topspin.
The bounce angle and speed of a ball sometimes depends on the amount of spin of the incident ball, but sometimes it doesn’t. It depends on the angle of incidence. Incident spin affects the bounce angle and speed only if the ball is incident at or near a right angle or within about 60 degrees to a right angle. If the ball is incident at a grazing angle (ie more than 60 degrees away from a right angle) then the bounce angle and speed will not depend on how fast the ball is spinning before it bounces.
FForces on the ball
When the ball hits the surface it starts to slide along the surface and it starts to squash. The bottom of the ball (green dot) slows down in the horizontal direction due to the friction force, F, but the top of the ball (blue dot) does not slow down. The ball therefore rotates on the surface, like a person tripping on a step, and it continues to rotate with topspin after it bounces. The ground reaction force N acts up on the ball causing it to slow down in the vertical direction until the whole ball comes to a stop in the vertical direction, then pushes the ball back up off the surface. N is equal and opposite to the vertical force of the ball pushing down on the surface.
If q is less than about 300 then the ball will keep sliding until it bounces. If q is greater than about 300 then the bottom of the ball will come to a complete stop in the horizontal direction, like the foot of a person walking along the surface. The surface then grips the ball (and vice versa) but the ball keeps rotating, causing the ball to twist out of shape, and causing F to reverse direction. After the ball starts to rise, N starts to drop, and the surface can no longer grip the ball. The bottom of the ball then slides backward on the surface due to its twist.
N usually acts slightly ahead of the centre of the ball since the front edge pushes down more firmly than the back edge (because the front edge rotates into the surface while the back edge rotates off the surface).
Superball contact time = 4 ms
Basketball contact time = 15 ms
Baseball contact time = 2.5 ms
Tennis ball contact time = 6 ms
2 p rBall spin
If a ball of radius r rolls along a surface at speed v, it travels a distance = 2p r as it rotates once about its axis, in time T. Hence v = 2p r / T. The time for one revolution is therefore T = 2p r / v. The number of revolutions per second is f = 1/T = v/ 2p r
For example, if r = 0.033 m (as it is for a tennis ball) and v = 20 m/s then f = 96 revolutions/sec = 5788 rpm (revolutions per minute). A similar thing happens when a ball bounces at speed v. That is, if a tennis ball bounces at a horizontal speed of 20 m/s then it will be spinning at about 5800 rpm. A ball doesn’t roll during a bounce since it grips or slides along the surface, but the rate of spin will be similar to that of a rolling ball. If the ball grips the surface when it bounces then it will rotate a bit faster than a rolling ball. If the ball slides throughout the bounce then it will rotate a bit slower than a rolling ball.
The best way to measure the bounce properties of various balls and surfaces is to film the bounce with a digital video camera and then transfer selected clips to a computer to analyse each frame. To measure ball spin, you can draw a line around a circumference with a felt tip pen and throw or project the ball with the line facing the camera along a diameter of the ball. Add a dot on one side of the line so you can tell how far the ball has rotated (using a protractor to measure its rotation from one frame to the next).
Some questions to resolve:
At what angle of incidence does the ball slow down the most and at what angle does it spin the fastest? Does it depend on the surface?
Does the COR vary with the angle of incidence? COR here is defined as the ratio of rebound speed in the vertical direction to the incident speed in the vertical direction ie vy(out)/vy(in) where y is the vertical direction.
Does vx(out)/vx(in) vary with angle of incidence? x being the horizontal direction.
Does the spin of the ball agree with the expected result that it will be similar to that for a rolling ball? Do small balls spin faster than big balls? Superballs spin a lot faster than other balls of similar size since they store and release elastic energy more efficiently, in directions both parallel and perpendicular to the surface on which they bounce. It is the un-twisting after the grip phase that allows them to spin so fast. They would make great golf balls if it was allowed by the rules since backspin allows a golf ball to travel further.