Bouncing Balls

1
Bouncing Balls

• Rod Cross
• Physics Department
• Sydney University
• June 2006

2
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
3
Vertical bounce
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
4
Vertical bounce experiments
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 doesnt matter
which, provided you are consistent. What problems
do you encounter? eg hard to pick the actual
bounce height, ball is not round and doesnt
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 doesnt 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
doesnt 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.
5
Contact time
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 cant 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 wont 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.
6
Bounce in the horizontal direction

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 20or
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 doesnt. 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.
7
Forces 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).
8
Bounce of various balls at low speed
Superball contact time 4 ms
Basketball contact time 15 ms
Baseball contact time 2.5 ms
Tennis ball contact time 6 ms
9
Ball 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 doesnt 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.
10
Ball bounce experiments

• 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.