Re: How does the temperature of a squash ball affect the height of bounce?

Greetings:

The answer to your question is very complex and scientists and chemists 
must resort to empirical models and measured materials properties to 
develop elastic materials with specific properties. I’ll try to explain 
this empirical process in my answer. Scientists could formulate a detailed 
analysis for each ball material at the atomic and molecular level ; 
however, the cost and time to do this work for each ball design would be 
prohibitive and the empirical methods offer the best practical solution to 
the problem. We begin our answer by finding what properties a standard 
squash ball must have. Note the specified temperature characteristics.

The World Squash Federation standard, which can be found at the following 
URL, states that squash balls have the following specification:

http://www.squash.org/WSF/rules.html#a7

(BEGIN QUOTATION) APPENDIX 7 
SPECIFICATIONS OF A STANDARD YELLOW DOT SQUASH BALL

The following specification is the standard for a yellow dot ball to be 
used under the Rules of Squash.

      Diameter (millimetres) 40.0 + or - 0.5
      Weight (grams) 24.0 + or - 1.0
      Stiffness (N/mm) @ 23 degrees C. 3.2 + or - 0.4
      Rebound Resilience - from 100 inches/254 centimetres 
              @ 23 degrees C. 12% minimum
              @ 45 degrees C. 26% - 33%
NOTES
      1. The full procedure for testing balls to the above specifications 
          is available from the WSF.
      2. No specifications are set for faster or slower speeds of ball, 
          which may be used by players of greater or lesser ability or in
          court conditions which are hotter or colder than
          those used to determine the yellow dot specification. Where         
          faster speeds of ball are produced they should bear the          
          following colour codes in ascending order of speed:
      Super slow - Yellow Dot
      Slow - White Dot or Green Dot
      Medium - Red Dot
      Fast - Blue Dot

      3. Yellow dot balls which are used at World Championships or at 
similar standards of play must meet the above specifications but additional 
subjective testing will be carried out by the WSF with players of the 
identified standard to determine the suitability of the nominated ball for 
Championship usage.”

(END QUOTATION)

The rebound resilience for a ball is related to a scientific term called 
elasticity. Elasticity deals with the behavior of those substances which 
have the property of recovering their size and shape when the forces 
producing deformations are removed. The elasticity is related to the stress 
and strain within the ball material  when it is under compression and 
elongation (Hooke’s Law) and a detailed analysis is mathmatically complex.   
Scientists have found that they can use a simple model to help understand 
and create deformable materials with specific properties.

Scientists model the atoms in an elastic material  as masses interconnected 
by an array of springs (chemical bonds) in a complex vibrating structure. 
The springs are always  vibrating (jiggling) and we call this vibration 
heat. However, the interconnected springs hold the total material in an 
overall fixed size and shape. To give you a size comparison, if a squash 
ball is magnified to the size of the earth the atoms would be about the 
size of a squash ball! 

The higher the temperature of  the substance the greater the vibration of 
the springs and if the temperature becomes hot enough the material melts or 
leaves as a gas and the springs are broken and the material is no longer 
elastic but becomes a fluid or gas. Also many elastic materials such as 
rubber balls become glass like at very cold temperatures and loose their 
elasticity and will shatter like glass under impact. Placing rubber balls 
in liquid nitrogen (-160 degrees C) and that shattering them is a old 
laboratory demonstration trick.  This tells us that the behavior of the 
springs holding the masses (atoms) together have a large temperature 
dependence. 

If a deformed material returns to its original shape after external forces 
are removed the material  is completely elastic. If the material remains in 
its deformed state after external forces are removed, such as a ball made 
of dough or wet clay hitting a wall,  the material is completely plastic. 
Real world deformable materials fall between these two extremes. To model 
real world materials  scientists also add  dashpots in parallel with the 
springs in their model. Dashpots are similar to the shock absorbers used in 
parallel with the springs in motor vehicles and they momentarily resist the 
change in elongation or compression of the springs.. 

When a ball hits a hard surface the kinetic energy of the ball’s motion 
rapidly compresses the springs in the direction of motion while the springs 
parallel to the wall are rapidly stretched. When the ball reaches zero 
velocity we have the maximum compression and stretching of the springs and 
the total energy in the ball is momentarily held in the elastic energy in 
the springs. The springs then rapidly return to their normal state 
converting the elastic energy back to kinetic energy. If the elasticity of 
the ball material was perfect, all of the energy in the springs would be 
converted to kinetic energy and the ball would leave the wall at the same 
velocity as it hit the wall. However, no spring is perfect and the dashpots 
(which are internal to the springs) are heated while being compressed and 
stretched. and the law of conservation of energy requires that  the heat 
energy added to the  dashpots/springs must come from the ball’s total 
kinetic energy of motion. This loss of energy results in the ball loosing 
velocity during each rebound which in turn heats the ball to a higher 
temperature after each rebound.  The springs and dashpots have a positive 
temperature coefficient which means the resilience increases with 
temperature and the balls become more lively, at least until the springs 
fly apart (fracture/melt or vaporize in the limit). For example if a yellow 
dot squash ball is dropped from the standard 100 inches (254 cm) it will 
hit the floor at 15.7 miles/hour (25 km/hr); however, it will rebound to 12 
inches (30.5 cm) starting from the floor at only 5.5 miles/hour (8.8 km/
hr). The 10 miles per hour (16km/hr) of velocity lost during the rebound is 
converted to heat in the ball.

In summary the answer to your question is very complex at the molecular 
level; however, springs and masses are used by scientists to model the 
material properties.  The springs in model elastic materials are highly 
temperature dependent and the resilience is determined by the very complex 
properties of the chemical bonds between the atoms and molecules in the 
ball material. The manufacturer of the ball must modify the elasticity of 
the material by combining and forming  materials with different spring 
constants and dashpot restraints (which are related to the nature of the 
chemical bonds) to meet the specific conditions  set by the rules of the 
game. Also, the rules state that the resilience of the ball can change by 
about 1% per degree C , with a positive temperature coefficient, during 
play as the ball temperature changes. 

While this answer may not be very satisfying what it tells us is that much 
of organic chemistry still relies on empirical models and formulas which 
often are in the form of patents and trade secrets!

Best regards, your Mad Scientist
Adrian Popa