The coefficient of restitution (COR) is a measure of the fraction of kinetic energy the objects will possess after they collide with one another compared to before they collide. The COR is a number between 0 and 1, where:
A COR of 1 is an elastic collision, in which there has been no loss of kinetic energy.
A COR of 0 is an entirely inelastic collision, in which the bodies become stuck together and lose the greatest kinetic energy.
Knowing the COR is helpful in the study and forecasting of objects’ action when they strike each other.
What is the Coefficient of Restitution?
The coefficient of restitution (COR, also denoted by e), is the ratio of the final to the initial relative speed between two objects after they collide. Another way of saying this is that the coefficient of restitution is the ratio of the velocity components along the normal plane of contact after and before the collision.
It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectly inelastic collision has a coefficient of 0, but a 0 value does not have to be perfectly inelastic. It is measured in the Leeb rebound hardness test, expressed as 1000 times the COR, but it is only a valid COR for the test, not as a universal COR for the material being tested.
The value is almost always less than 1 due to initial translational kinetic energy being lost to rotational kinetic energy, plastic deformation, and heat. It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity.
Coefficient of restitution: A simple explanation
When two objects collide with each other, many forces come into play, which also means the application of various mathematical equations. Many of these laws were first derived by the same super popular scientist who is credited with numerous discoveries and derivations, meaning that he has a number of patents to his name – Sir Isaac Newton.
Pertaining to the collision of two objects, Newton formulated a theory that we now know as Newton’s law of restitution. It simply states that when two bodies collide, the speed with which they move after the collision depends on the material from which they are made.
Let’s suppose a rubber ball bounces on a flat, hard surface. Obviously, the rubber ball will rebound off the surface, but with only a fraction of its original energy, because all real collisions are inelastic. (Note: If this collision were elastic, then the ball would have bounced back with the same amount of energy it had before striking the surface.)
You see, when you ‘deform’ something by colliding it with something else (say, when you bounce a basketball on the ground), a fraction of its original energy is lost. That’s why the basketball bounces lower with every collision – as its energy gets converted to heat/vibrations.
In this case, you can think of the coefficient of restitution as an entity that tells you how efficient the “bouncing” process is. The more efficient it is, the ‘bouncier’ the basketball shall be.
Coefficient of Restitution Formula
The mathematical formula of the Coefficient of Restitution is given as follows:
The coefficient of restitution was developed by Sir Isaac Newton in 1687. It is also known as Newton’s experimental law.
From the above equation, you notice that you always divide the smaller number by a larger number. Therefore, the coefficient of restitution is always positive.
The value is almost always less than one due to initial translational kinetic energy being transformed to rotational kinetic energy, plastic deformation, and heat. However, it can be more than one if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity.
Range of Values for e
- If e = 0, then it is a perfectly inelastic collision
- If 0 < e < 1, then it is a real-world inelastic collision, in which some kinetic energy is dissipated.
- If e = 1, then it is a perfectly elastic collision in which no kinetic energy is dissipated, and the objects rebound from one another with the same relative speed with which they approached.
How to Calculate Coefficient of Restitution
COR can be calculated using the equation:
e = (Relative velocity after collision)/(Relative velocity before collision)
For two objects colliding with each other (object 1 and object 2):
e = (v2′ – v1′)/(v1 – v2)
Note: v1 and v2 are the speeds of the two bodies before collision, and v1′ and v2 are their speeds after collision.
Factors Affecting the Coefficient of Restitution
1. Material Properties
Different materials have varying inherent elasticity, which directly affects their COR. For example:
Rubber ball: COR ≈ 0.85–0.95, such that it will bounce almost back to its initial height when dropped.
Steel ball against steel: COR ≈ 0.6–0.7. Steel is stiffer than rubber, so more kinetic energy is dissipated in the collision.
This is why balls at a playground bounce higher than metal bearings dropped from the same height.
2. Surface Texture
Smoothness or roughness of surfaces colliding plays a decisive role:
Smooth glass-on-glass collision: COR ≈ 0.9, very little energy lost to friction.
Rough steel-concrete impact: COR ≈ 0.4–0.5, since surface roughness and micro-asperities lose energy.
Elastic collision with smooth surfaces, while rough or deformed surfaces lose energy through vibration and friction.
3. Collision Velocity
Collision velocity can alter the COR, particularly in those materials that are strain-rate sensitive:
Low-velocity drop of a steel ball (1 m/s): COR ≈ 0.65
High-strain rate steel ball impact (10 m/s): COR can reduce to ≈ 0.55 due to microplastic deformation at high strain rates.
Influence is particularly critical in high-speed applications such as ballistics or crash testing of vehicles.
4. Temperature
Temperature affects the elasticity of the material and thus the COR:
Rubber at 20°C: COR ≈ 0.9
Rubber at 0°C: COR reduces to ≈ 0.75 due to the stiffness of the material
Metal alloys: Some metals have lower COR at very low temperatures due to increased brittleness.
Temperature-dependent COR is of particular interest to outdoor sport, aerospace, and cryogenics.
Applications of the Coefficient of Restitution
COR is a key factor in design and analysis of collision systems. Accurate knowledge of COR allows designers and engineers to predict energy transfer, optimize performance, and enhance safety in various applications.
1. Sports Equipment Design
COR directly impacts behavior and ball bounce on play in sports. For instance:
Basketballs: A reading of approximately 0.75 COR when dropped from a height of 1.8 meters is found in an NBA regulation basketball to ensure consistent bounce on hardwood floors.
Tennis balls: COR between 0.55 and 0.60 depending on surface type and temperature, which in turn affects rebound speed and control.
These properties are also utilized by producers to modify the composition of balls, air pressure, and coatings for peak performance.
2. Automobile Safety
COR is utilized in car design to model collisions and design safety elements. For example:
Crumple zones: With knowledge of the COR of different metals and composites in car frames (steel ≈ 0.6, aluminum ≈ 0.55), designers can estimate energy absorption in impacts and reduce transmitted forces to passengers.
Crash testing: Automobile bumpers and crash barriers are tested for COR to predict rebound after impact and minimize secondary collisions.
3. Robotics and Automation
Contacting robots must know actual COR values in order to control motion and avoid damage:
Industrial manipulators: During the movement of metal or plastic parts across assembly lines, COR values (metal-on-metal ≈ 0.6–0.7, plastic-on-metal ≈ 0.5) are used in simulations to predict post-collision motion.
Service robots: In ball-sorting or object-delivery applications, accurate measurement of COR ensures sufficient forces by the robot to avoid objects from bouncing beyond reach.
Through the application of COR in design and analysis, designers have the ability to optimize performance, enhance safety, and make better predictions for the performance of sports, transportation, and automation systems.