Zenit Research Lab · Brown University · 2025
Sport Mechanics: The Influence of the Coefficient of Restitution on the Agility to Play Squash
- Role
- Undergraduate research assistant
- Supervisor
- Professor Roberto Zenit
Abstract
The purpose of this project was to understand the relationship between the coefficients of restitution (CoRs) for squash balls with different physical features under different conditions. We calculated CoRs for squash balls of different ratings and ages at different starting temperatures fired using an airgun at various pressure levels. Images of the ball trajectory were captured using a high-speed camera which allowed us to compute the pre- and post-impact velocity of the balls. Preliminary findings indicate that temperature plays a strongly significant and positive role in the CoR. Furthermore, an increase in initial velocity seems to cause the CoR to decrease, and this effect is more pronounced at a higher temperature. It initially appeared that double-dot balls had lower CoRs than single-dot ones, but the effect of ball type on CoR was statistically insignificant. It also initially appeared that old balls unexpectedly had higher CoRs than new balls; however, this effect was also insignificant.
§ 01
Introduction
Squash is played in an indoor court, where two players hit a ball against the front wall and can use the side and back walls. The ball may bounce on the floor once per shot, only after it has hit the front wall. Players can select different ball types, usually depending on their level. Conventional wisdom holds that the bouncier the ball, the easier it is to play with, because it allows more time to react and move.
The single yellow dot ball is a competition ball, which bounces more than the double yellow dot professional ball. Players often heat balls before matches to increase bounce. An old ball typically shows wear, becomes softer, and is considered less bouncy than a new one.
Squash is a high-intensity sport, and efficient court movement is critical at the highest levels. Players ideally strike their shots and return to the T at the centre of the court. If the ball bounces less, a player has less time to recover position and is less likely to reach the next shot.
The objective of this research was to understand the relationship between coefficients of restitution and the physical attributes of squash balls, helping players select ball type, age, and temperature with clearer evidence. Given pre- and post-impact velocities for each ball and condition, a coefficient of restitution (CoR) can be calculated to measure bounciness under each circumstance.
§ 02
Hypotheses
Temperature
A warmer ball would have a higher CoR, because internal air pressure increases and the rubber becomes more elastic.
Age
An older ball would have a lower CoR, because repeated high-velocity impacts cause microcracking and permanent deformation in the rubber.
Ball level
A single-dot ball would have a higher CoR than a double-dot ball, which uses denser rubber to bounce less.
Initial velocity
No initial expectation of whether impact velocity would increase or decrease the CoR.
§ 03
Materials & method
The experiment
An airgun was used in a lab to fire balls into a wall clamped to a desk surface. A 1 cm × 1 cm grid was placed behind the wall to track motion and provide a distance reference. Collisions were recorded under bright lighting with a high-speed camera at 4000 fps.
Four squash balls were tested: two single-dot and two double-dot, including one new and one old ball of each type. Each ball was fired at room temperature (22 °C) and when heated (35 °C) using an electric heating blanket; temperature was recorded with a laser thermometer. For each ball and temperature combination, the airgun was filled to 5, 7.5, and 10 psi to vary initial velocity.
High-speed images were used to calculate pre- and post-impact velocity. A MATLAB script tracked the x-position of the ball over time, producing calculations for initial velocity, final velocity, and CoR. The script accepted an 8 cm reference to convert pixel distance into real distance.
Coefficient of restitution
e = |v₂ − v₁| / |u₂ − u₁|
With a stationary wall, this reduces to e = v / u, where u is pre-impact velocity and v is post-impact velocity. A CoR of 1 would mean no energy loss; values lie between 0 and 1. CoRs were calculated across 24 ball type / temperature / age / velocity combinations.
§ 04
Results
Temperature & initial velocity
Temperature plays a significant and positive role in the value of e. An increase in initial velocity appears to cause e to decrease, and this effect is more pronounced at 35 °C than at 22 °C.
Temperature & ball type
At first glance, double-dot balls appear to have somewhat lower average CoR. A two-tailed, paired-dependent t-test between averages at each temperature showed neither difference was significant. Results for ball type may be unreliable given confounding variables and only six observations per measurement. For each ball type, however, higher temperature consistently produced higher CoRs; t-tests on temperature differences were highly significant.
| Temperature (°C) | 1-dot CoR | 2-dot CoR | Type p-value |
|---|---|---|---|
| 22 | 0.291 | 0.262 | 0.255 |
| 35 | 0.394 | 0.376 | 0.715 |
Temperature difference p-values: 0.038* (1-dot) and 0.017* (2-dot). *Significant at 5%.
22 °C
35 °C
Temperature & ball age
Old balls initially appeared to have slightly higher e, but the difference was statistically insignificant; temperature remained the dominant factor. Paired t-tests between old and new averages at each temperature were not significant, likely for the same confounding and sample-size reasons as ball type.
| Temperature (°C) | Old balls CoR | New balls CoR | Age p-value |
|---|---|---|---|
| 22 | 0.291 | 0.263 | 0.271 |
| 35 | 0.395 | 0.375 | 0.686 |
Temperature difference p-values: 0.045* (old) and 0.011** (new). *Significant at 5%; **significant at 1%.
22 °C
35 °C
§ 05
Discussion
CoR as a function of initial velocity decreases, and this effect is more prominent at 35 °C than at 22 °C. Across all analyses, temperature is the greatest factor: at a given velocity, higher temperatures consistently yield higher e, consistent with the initial hypothesis.
Ball type and age show small apparent differences that are statistically insignificant, surprising given how much players emphasise both factors. These findings should be treated cautiously because several variables changed when producing averages and sample sizes were small.
The next step is to repeat the experiment many more times with tighter control of confounding variables, more temperature and velocity observations, and regression analysis to estimate CoRs while controlling for all factors simultaneously.
§ 06
Conclusion
The coefficient of restitution was strongly dependent on temperature, with significant increases in CoR between 22 °C and 35 °C across all conditions. Ball age and ball type did not show significant effects. CoR also decreased as impact velocity increased, though this relationship was secondary to temperature.
Players' habit of warming balls appears grounded in genuine physical results. The advantage of hitting the ball hard is real in most situations, abstracting from strategic plays like drop shots. This research casts doubt on whether ball level or age materially affects difficulty of play.
§ 07
Next stage
The next stage is determining how CoR affects player agility in match conditions. Matches will be recorded with players using balls of known CoR, varying ball type and temperature, using a wide-angle camera mounted on the back wall. Player motion can then be tracked to measure average speed and total distance travelled, indicating whether players move more and rallies last longer with bouncier balls. Software such as smartsquash.ai may support this analysis when players wear distinguishable clothing.