Coefficient of restitution

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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.

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 .83. Golf balls also have a COR of about .78^[1]. 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."^[2].

1 Equation
2 Further details
3 Use
- 3.1 Derivation
4 See also
5 References

[edit] Equation

The coefficient itself is given by:

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

for two colliding objects, where

V 1 f

is the scalar final velocity of the first object after impact

V 2 f

is the scalar final velocity of the second object after impact

V 1

is the scalar initial velocity of the first object before impact

V 2

is the scalar initial velocity of the second object before impact

and:

$C_R = {\frac{V_{f}}{V_{i}}}$

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

V f

is the scalar velocity of the object after impact

V i

is the scalar velocity of the object before impact

The coefficient can also be found with:

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

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

h

is the bounce height

H

is the drop height

Note that the velocities used are scalar products pointing along the direction of impact (or perpendicular to the tangental vector of the point of contact, if you prefer). For spheres, this direction would be $p o s 2 - p o s 1$ normalized, where $p o s x$ is the position of the xth sphere. A simple dot product can then be used between this vector and the velocity vector to find the scalars.

[edit] Further details

The COR is generally a number in the range [0,1]. 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. 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. If you had 5 different types of objects colliding, you would have (5 choose 2 = 10) 10 different CORs, one for each possible collision between any two object types.

[edit] 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.

$V_{1f} = \frac{ (c + 1)M_{2}V_{2} + V_{1}(M_{1} - cM_{2})}{M_{1} + M_{2}}$

and

$V_{2f} = \frac{ (c + 1)M_{1}V_{1} + V_{2}(M_{2} - cM_{1})}{M_{1} + M_{2}}$

where

V 1 f

is the final velocity of the first object after impact

V 2 f

is the final velocity of the second object after impact

V 1

is the initial velocity of the first object before impact

V 2

is the initial velocity of the second object before impact

M 1

is the mass of the first object

M 2

is the mass of the second object

c

is the coefficient of restitution

[edit] 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):

$\begin{cases} M_{1}V_{1f} + M_{2}V_{2f} = (M_{1}V_{1} + M_{2}V_{2})\\ -V_{1f} + V_{2f} = c(V_{2} - V_{1}) \end{cases}$