Definitions

# Coefficient of restitution

The coefficient of restitution or COR of an object is a fractional value representing the ratio of velocities before and after an impact. An object with a COR of 1 collides elastically, while an object with a COR of 0 will collide inelastically, effectively "sticking" to the object it collides with, not bouncing at all.

## Common usage

The coefficient of restitution entered the common vocabulary, among golfers at least, when golf club manufacturers began making thin-faced drivers with a so-called "trampoline effect" that creates drives of a greater distance as a result of an extra bounce off the clubface. The USGA (America's governing golfing body) has started testing drivers for COR and has placed the upper limit at 0.83. Golf balls also have a COR of about 0.78. According to one article (addressing COR in tennis racquets), "[f]or 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.

The International Table Tennis Federation specifies that the ball must have a coefficient of restitution of 0.94.

## Equation

Consider a one-dimensional collision. Velocity in one direction is labeled "positive" and the opposite direction "negative".

The coefficient of restitution is given by:

$C_R = frac\left\{V_\left\{2f\right\} - V_\left\{1f\right\}\right\}\left\{V_\left\{1\right\} - V_\left\{2\right\}\right\}$

for two colliding objects, where

$V_\left\{1f\right\}$ is the scalar final velocity of the first object after impact
$V_\left\{2f\right\}$ is the scalar final velocity of the second object after impact
$V_\left\{1\right\}$ is the scalar initial velocity of the first object before impact
$V_\left\{2\right\}$ is the scalar initial velocity of the second object before impact

Even though the equation does not reference mass, it is important to note that it still relates to momentum since the final velocities are dependent on mass.

For an object bouncing off a stationary object, such as a floor:

$C_R = -frac\left\{V_\left\{f\right\}\right\}\left\{V_\left\{i\right\}\right\}$, where

$V_\left\{f\right\}$ is the scalar velocity of the object after impact
$V_\left\{i\right\}$ is the scalar velocity of the object before impact

The coefficient can also be found with:

$C_R = sqrt\left\{frac\left\{h\right\}\left\{H\right\}\right\}$

for an object bouncing off a stationary object, such as a floor, where

$h$ is the bounce height
$H$ is the drop height

For two- and three-dimensional collisions the velocities used are the components perpendicular to the tangent line/plane at the point of contact.

## Further details

The COR is generally a number in the range [0,1]. Qualitatively, 1 represents a perfectly elastic collision, while 0 represents a perfectly inelastic collision. A COR greater than one is theoretically possible, representing a collision that generates kinetic energy, such as land mines being thrown together and exploding. For other examples, some recent studies have clarified that COR can take a value greater than one in a special case of oblique collisions These phenomena are due to the change of rebound trajectory of a ball caused by a soft target wall. A COR less than zero is also theoretically possible, representing a collision that pulls two objects closer together instead of bouncing them apart.

An important point: the COR is a property of a collision, not necessarily an object. For example, if you had 5 different types of objects colliding, you would have $\left\{5 choose 2\right\} = 10$ different CORs (ignoring the possible ways and orientations in which the objects collide), one for each possible collision between any two object types.

## Use

The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions as well, and every possibility in between.

$V_\left\{1f\right\}=frac\left\{\left(C_R + 1\right)M_\left\{2\right\}V_2+V_\left\{1\right\}\left(M_1-C_R M_2\right)\right\}\left\{M_1+M_2\right\}$
and
$V_\left\{2f\right\}=frac\left\{\left(C_R + 1\right)M_\left\{1\right\}V_1+V_\left\{2\right\}\left(M_2-C_R M_1\right)\right\}\left\{M_1+ M_2\right\}$

where

$V_\left\{1f\right\}$ is the final velocity of the first object after impact
$V_\left\{2f\right\}$ is the final velocity of the second object after impact
$V_\left\{1\right\}$ is the initial velocity of the first object before impact
$V_\left\{2\right\}$ is the initial velocity of the second object before impact
$M_\left\{1\right\}$ is the mass of the first object
$M_\left\{2\right\}$ is the mass of the second object

### Derivation

The above equation can be derived from the analytical solution to the system of equations generated by the definition of the COR and the law of the conservation of momentum (which holds for all collisions):


begin{cases} M_{1}V_{1f} + M_{2}V_{2f} = (M_{1}V_{1} + M_{2}V_{2}) -V_{1f} + V_{2f} = C_R(V_{1} - V_{2}) end{cases}