Coefficient of restitution

The Coefficient of Restitution (e) serves as a vital numerical bridge between the idealized world of physics and the complex reality of physical impacts, acting as a measure of how much "bounce" remains in a system after two objects collide. At its core, this parameter represents the ratio of the relative speeds of the objects before and after they touch, providing a clear indication of how much kinetic energy is preserved versus how much is lost to the environment. When an object strikes a surface and rebounds with significant vigor, it possesses a high coefficient, whereas a soft or malleable object that thuds and stays put has a coefficient approaching the bottom of the scale. This value is never truly a fixed constant of a single material, but rather a dynamic result of the interaction between two specific surfaces, such as a rubber tire meeting asphalt or a steel hammer hitting an anvil. Because real-world materials are rarely perfectly resilient, this coefficient almost always falls within a specific range that accounts for the energy dissipated as heat, vibrational noise, or permanent structural changes like dents and cracks. Designers and engineers across various industries rely on this metric to predict exactly how components will behave during high-stakes impacts, ensuring that everything from protective sports gear to automotive safety systems performs with calculated precision. By quantifying the efficiency of a rebound, the coefficient allows us to move beyond simple guesswork and mathematically describe the subtle differences between a sharp, elastic snap and a dull, energy-absorbing impact.

The physical implications of this concept are most visible in the way energy is transformed during the split second of contact when objects compress and then attempt to return to their original shapes. A perfectly elastic interaction would suggest that no energy is wasted, yet in our physical universe, the microscopic friction between molecules during compression ensures that some energy is always diverted away from the rebound motion. In the context of manufacturing and quality control, measuring this coefficient is a standard way to test the structural integrity and performance consistency of consumer products like tennis balls, golf clubs, and even smartphone screens. For instance, a high coefficient in a sports ball ensures a consistent and predictable trajectory for athletes, while a low coefficient in a bumper system is actually preferred because it indicates the material is successfully absorbing a dangerous impact. The behavior of this value can also change depending on environmental conditions; for example, a cold basketball may feel "dead" because its internal pressure and material flexibility have dropped, resulting in a lower coefficient of restitution compared to a warm ball. Understanding these nuances is essential for physicists who model the chaotic movements of particles in a gas or the grand-scale collisions of celestial bodies in space. Ultimately, the coefficient of restitution provides the necessary context to explain why some things spring back to life after a fall while others succumb to the force of gravity and come to a complete, motionless stop.

Coefficient of restitution
A bouncing ball captured with a stroboscopic flash. Ignoring air resistance, the square root of the ratio of the height of one bounce to that of the preceding bounce gives the coefficient of restitution for the ball/surface impact
Other names	COR
Common symbols	e
SI unit	unitless

In physics, the coefficient of restitution (COR, also denoted by e), can be thought of as a measure of the elasticity of a collision between two bodies. It is a dimensionless parameter defined as the ratio of the relative velocity of separation after a two-body collision to the relative velocity of approach before collision. In most real-world collisions, the value of e lies somewhere between 0 and 1, where 1 represents a perfectly elastic collision (in which the objects rebound with no loss of speed but in the opposite directions) and 0 a perfectly inelastic collision (in which the objects do not rebound at all, and end up touching). The basic equation, sometimes known as Newton's restitution equation, was developed by Sir Isaac Newton in 1687. ^[1] $${\displaystyle {\text{Coefficient of restitution }}(e)={\frac {\left|{\text{Relative velocity of separation after collision}}\right|}{\left|{\text{Relative velocity of approach before collision}}\right|}}}$$

Introduction

As a property of paired objects

The COR is a property of a pair of objects in a collision, not a single object. If a given object collides with two different objects, each collision has its own COR. When a single object is described as having a given coefficient of restitution, as if it were an intrinsic property without reference to a second object, some assumptions have been made – for example that the collision is with another identical object, or with a perfectly rigid wall.

Treated as a constant

In a basic analysis of collisions, e is generally treated as a dimensionless constant, independent of the mass and relative velocities of the two objects, with the collision being treated as effectively instantaneous. An example often used for teaching is the collision of two idealised billiard balls. Real world interactions may be more complicated, for example where the internal structure of the objects needs to be taken into account, or where there are more complex effects happening during the time between initial contact and final separation.

Range of values for e

e is usually a positive, real number between 0 and 1:

• e = 0: This is a perfectly inelastic collision in which the objects do not rebound at all and end up touching.
• 0 < e < 1: This is a real-world inelastic collision, in which some kinetic energy is dissipated. The objects rebound with a lower separation speed than the speed of approach.
• e = 1: This is a perfectly elastic collision, in which no kinetic energy is dissipated. The objects rebound with the same relative speed with which they approached.

Values outside that range are in principle possible, though in practice they would not normally be analysed with a basic analysis that takes e to be a constant:

• e < 0: A COR less than zero implies a collision in which the objects pass through one another, for example a bullet passing through a target.
• e > 1: This implies a superelastic collision in which the objects rebound with a greater relative speed than the speed of approach, due to some additional stored energy being released during the collision.

Equations

In the case of a one-dimensional collision involving two idealised objects, A and B, the coefficient of restitution is given by:$${\displaystyle e={\frac {\left|v_{\text{b}}-v_{\text{a}}\right|}{\left|u_{\text{a}}-u_{\text{b}}\right|}},}$$ where:

• ${\displaystyle v_{\text{a}}}$ is the final velocity of object A after impact
• ${\displaystyle v_{\text{b}}}$ is the final velocity of object B after impact
• ${\displaystyle u_{\text{a}}}$ is the initial velocity of object A before impact
• ${\displaystyle u_{\text{b}}}$ is the initial velocity of object B before impact

This is sometimes known as the restitution equation. For a perfectly elastic collision, e = 1 and the objects rebound with the same relative speed with which they approached. For a perfectly inelastic collision e = 0 and the objects do not rebound at all.

For an object bouncing off a stationary target, e is defined as the ratio of the object's rebound speed after the impact to that prior to impact:$${\displaystyle e={\frac {v}{u}},}$$ where

• ${\displaystyle u}$ is the speed of the object before impact
• ${\displaystyle v}$ is the speed of the rebounding object (in the opposite direction) after impact

In a case where frictional forces can be neglected and the object is dropped from rest onto a horizontal surface, this is equivalent to:$${\displaystyle e={\sqrt {\frac {h}{H}}},}$$ where

• ${\displaystyle H}$ is the drop height
• ${\displaystyle h}$ is the bounce height

The coefficient of restitution can be thought of as a measure of the extent to which energy is conserved when an object bounces off a surface. In the case of an object bouncing off a stationary target, the change in gravitational potential energy, E_p, during the course of the impact is essentially zero; thus, e is a comparison between the kinetic energy, E_k, of the object immediately before impact with that immediately after impact:$${\displaystyle e={\sqrt {\frac {E_{\text{k, (after impact)}}}{E_{\text{k, (before impact)}}}}}={\sqrt {\frac {{\frac {1}{2}}mv^{2}}{{\frac {1}{2}}mu^{2}}}}={\sqrt {\frac {v^{2}}{u^{2}}}}={\frac {v}{u}}}$$In a case where frictional forces can be neglected (nearly every student laboratory on this subject ^[2] ), and the object is dropped from rest onto a horizontal surface, the above is equivalent to a comparison between the E_p of the object at the drop height with that at the bounce height. In this case, the change in E_k is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: $${\displaystyle e={\sqrt {\frac {E_{\text{p, (at bounce height)}}}{E_{\text{p, (at drop height)}}}}}={\sqrt {\frac {mgh}{mgH}}}={\sqrt {\frac {h}{H}}}}$$

Velocity and energy after impact

Velocity

Although e does not vary with the masses of the colliding objects, their final velocities are mass-dependent due to conservation of momentum:$${\displaystyle v_{\text{a}}={\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{b}}e(u_{\text{b}}-u_{\text{a}})}{m_{\text{a}}+m_{\text{b}}}}}$$ and $${\displaystyle v_{\text{b}}={\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{a}}e(u_{\text{a}}-u_{\text{b}})}{m_{\text{a}}+m_{\text{b}}}}}$$ where

• ${\displaystyle v_{\text{a}}}$ is the velocity of A after impact
• ${\displaystyle v_{\text{b}}}$ is the velocity of B after impact
• ${\displaystyle u_{\text{a}}}$ is the velocity of A before impact
• ${\displaystyle u_{\text{b}}}$ is the velocity of B before impact
• ${\displaystyle m_{\text{a}}}$ is the mass of A
• ${\displaystyle m_{\text{b}}}$ is the mass of B

Energy

Kinetic energy loss in CM frame: $${\displaystyle \Delta T=-(1-e^{2})\cdot \Delta T_{0,{\text{CM}}}=-(1-e^{2})\cdot {\frac {1}{2}}\cdot {\frac {m_{\text{a}}m_{\text{b}}}{m_{\text{a}}+m_{\text{b}}}}\cdot (u_{\text{a}}-u_{\text{b}})^{2}}$$

where

• ${\displaystyle \Delta T}$ is the kinetic energy lost during the collision
• ${\displaystyle e}$ is the coefficient of restitution
• ${\displaystyle \Delta T_{0,{\text{CM}}}}$ is the initial kinetic energy in the center of mass frame
• ${\displaystyle m_{\text{a}}}$ is the mass of object A
• ${\displaystyle m_{\text{b}}}$ is the mass of object B
• ${\displaystyle u_{\text{a}}}$ is the velocity of A before collision
• ${\displaystyle u_{\text{b}}}$ is the velocity of B before collision

Practical issues

Measurement

In practical situations, the coefficient of restitution between two bodies may have to be determined experimentally, for example using the Leeb rebound hardness test. This uses a tip of tungsten carbide, one of the hardest substances available, dropped onto test samples from a specific height.

A comprehensive study of coefficients of restitution in dependence on material properties (elastic moduli, rheology), direction of impact, coefficient of friction and adhesive properties of impacting bodies can be found in Willert (2020). ^[3]

Application in sports

See also: Bouncing ball § Sport regulations

Thin-faced golf club drivers utilize a "trampoline effect" that creates drives of a greater distance as a result of the flexing and subsequent release of stored energy which imparts greater impulse to the ball. The USGA (America's governing golfing body) tests ^[4] drivers for COR and has placed the upper limit at 0.83. COR is a function of rates of clubhead speeds and diminish as clubhead speed increase. ^[5] In the report COR ranges from 0.845 for 90 mph to as low as 0.797 at 130 mph. The above-mentioned "trampoline effect" shows this since it reduces the rate of stress of the collision by increasing the time of the collision. According to one article (addressing COR in tennis racquets), "for the Benchmark Conditions, the coefficient of restitution used is 0.85 for all racquets, eliminating the variables of string tension and frame stiffness which could add or subtract from the coefficient of restitution." ^[6]

The International Table Tennis Federation specifies that the ball shall bounce up 24–26 cm when dropped from a height of 30.5 cm on to a standard steel block, ^[7] implying a COR of 0.887 to 0.923.

The International Basketball Federation (FIBA) rules require that the ball rebound to a height of between 1035 and 1085 mm when dropped from a height of 1800 mm, ^[8] implying a COR between 0.758 and 0.776.

See also

• Bouncing ball
• Collision
• Damping capacity
• Resilience

References

1. Weir, G.; McGavin, P. (8 May 2008). "The coefficient of restitution for the idealized impact of a spherical, nano-scale particle on a rigid plane". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 464 (2093): 1295–1307. Bibcode:2008RSPSA.464.1295W. doi:10.1098/rspa.2007.0289. S2CID   122562612.
2. Mohazzabi, Pirooz (2011). "When Does Air Resistance Become Significant in Free Fall?". The Physics Teacher. 49 (2): 89–90. Bibcode:2011PhTea..49...89M. doi:10.1119/1.3543580.
3. Willert, Emanuel (2020). Stoßprobleme in Physik, Technik und Medizin: Grundlagen und Anwendungen (in German). Springer Vieweg. doi:10.1007/978-3-662-60296-6. ISBN   978-3-662-60295-9. S2CID   212954456.
4. Conforming Golf Club usga.org Archived 16 June 2021 at the Wayback Machine
5. "Do Long Hitters Get An Unfair Advantage?". USGA. 14 February 2015. Retrieved 1 June 2023.
6. "Coefficient of Restitution". Archived from the original on 23 November 2016.
7. "Tennis Tech resources | ITF". Archived from the original on 3 December 2019.
8. "FIBA basketball". FIBA.basketball. Retrieved 17 October 2024. (See page 12 of the Official Basketball Rules 2024 - Basketball Equipment, a pdf document downloadable from the Equipment & Venue tab of FIBA.basketball, and available at https://assets.fiba.basketball/image/upload/documents-corporate-fiba-official-rules-2024-official-basketball-rules-and-basketball-equipment.pdf)

Works cited

• Cross, Rod (2006). "The bounce of a ball" (PDF). Physics Department, University of Sydney, Australia. Retrieved 16 January 2008.{{cite journal}}: Cite journal requires |journal= (help)
• Walker, Jearl (2011). Fundamentals Of Physics (9th ed.). David Halliday, Robert Resnick, Jearl Walker. ISBN   978-0-470-56473-8.

External links

• Wolfram Article on COR
• Bennett & Meepagala (2006). "Coefficients of Restitution". The Physics Factbook.
• Chris Hecker's physics introduction
• "Getting an extra bounce" by Chelsea Wald
• FIFA Quality Concepts for Footballs – Uniform Rebound
• Bowley, Roger (2009). "Coefficient of Restitution". Sixty Symbols. Brady Haran for the University of Nottingham.
• Brogliato, Bernard (2016). Nonsmooth Mechanics. Models, Dynamics, and Control. Communications and Control Eng. Springer Int. Pub. Switzerland. ISBN   978-3-319-28662-4.