# What is the coefficient of restitution

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

For a collision between two objects, the coefficient of restitution is the ratio of the relative speed after to the relative speed before the collision.

The coefficient of restitution is a number between 0 (perfectly inelastic collision) and 1 (elastic collision) inclusive.

Equations

The coefficient of restitution is

$$\textrm{C.O.R.} = \frac{|\vec{v_{2f}} - \vec{v_{1f}}|} {|\vec{v_{2i}} - \vec{v_{1i}}|}$$

where $\vec{v_{1i}}$ and $\vec{v_{1f}}$ are the initial and final velocities, respectively, of object #1. A similar definition holds for the velocities of object #2.

While this is a useful definition for studying collisions of particles in physics, there is an alternative used to define the C.O.R. of everyday objects. In this definition, the velocities are replaced with the components perpendicular to the plane or line of impact. In the case of a 1-d collision, the two definitions are equivalent.

Be sure you know which definition of C.O.R. is the accepted practice in a given situation. For the remainder of this discussion, we use the definition in the equation shown above.

For an object colliding with a fixed object or surface, $v_{2i}$ and $v_{2f}$ are zero, and the C.O.R reduces to:

$$\textrm{C.O.R.} = \frac{|\vec{v_{1f}}|}{|\vec{v_{1i}}|}$$

In the center-of-mass reference frame of two objects of mass $m_1$ and $m_2$ -- and only in that frame -- the initial and final total kinetic energies are related to the C.O.R. by

$$\frac{KE_f}{KE_i} = \textrm{C.O.R}^2$$

where

$$KE_i \ = \ \frac{1}{2} m_1 v_{1i}^2 \ + \ \frac{1}{2} m_2 v_{2i}^2$$

and

$$KE_f \ = \ \frac{1}{2} m_1 v_{1f}^2 \ + \ \frac{1}{2} m_2 v_{2f}^2$$

Extended explanation

Elastic and perfectly inelastic collisions

The coefficient of restitution describes the inelasticity of collisions. If the C.O.R. is 1, the collision is elastic and kinetic energy is conserved. A C.O.R of zero represents a perfectly inelastic collision; after the collision the objects stick together and, in the center-of-mass frame, have zero velocity.

Simplifying 1-d collision problems

In a 1-dimensional elastic collision (C.O.R. = 1), the conservation-of-energy equation may be replaced with

$$v_{2f} - v_{1f} = v_{1i} - v_{2i}$$

In other words, the relative velocity of the two particles has the same magnitude, but is reversed in direction, before and after the collision. By using this equation instead of the conservation-of-energy equation directly, the work of solving a collision problem is simplified as there are no squared velocity terms to deal with.

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2. Mar 23, 2015

### Abeerchughtai

What does coefficient of restitution signify? on what factor does it depends?

3. May 31, 2015

### G.H.Hobbs

The COR represents the elasticity of an impact between two bodies,the impact being either perfectly elastic or inelastic,or somewhere in between.In terms of the numbers simply look at the equation ; its the ratio of the difference of the velocities before and after a collision or impact has happened.In more advanced mechanics classes its use is applied in the energy equations,it comes in handy,when more has more unknowns than equations and wants a closed form solution without complicating one's life with life threatening PDE's.

4. Aug 24, 2015

5. May 9, 2016