What is the Coefficient of Restitution?

What is 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.)

Coefficient of Restitution

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:

Coefficient of Restitution Formula

Mathematics 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 more significant 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.