Coefficient of restitution
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Fraction (mathematics)
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al value representing the ratio of speeds after and before an impact, taken along the line of the impact. Pairs of objects with COR 1 collide elastically
Elastic collision
An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter...
, while objects with COR < 1 collide inelastically
Inelastic collision
An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved.In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.The molecules of a gas...
. For a COR = 0, the objects effectively "stop" at the collision, not bouncing at all. An object (singular) is often described as having a coefficient of restitution as if it were an intrinsic property without reference to a second object, in this case the definition is assumed to be with respect to collisions with a perfectly rigid and elastic object. COR = (relative speed after collision)/(relative speed before collision).
Further details
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 would represent a collision in which the separation velocity of the objects has the same direction (sign) as the closing velocity, implying the objects passed through one another without fully engaging. This may also be thought of as an incomplete transfer of momentum. An example of this might be a small, dense object passing through a large, less dense one - e.g. a bullet passing through a target, or a motorcycle passing through a motor home or a wave tearing through a dam.
An important point: the COR is of a collision, not necessarily an object. For example, if you had five different types of objects colliding, you would have
different CORs (ignoring the possible ways and orientations in which the objects collide), one for each possible collision between any two object types.Generally, the COR is thought to be independent of collision speed. However, in a series of experiments performed at Florida State University in 1955, it was shown that the COR varies as the collision speed approaches zero, first rising significantly as the speed drops, then dropping significantly as the speed drops to about 1 cm/s a again as the collision speed approaches zero. This effect was observed in slow speed collisions involving a number of different metals.
Sports equipment
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 typically have a COR of about 0.78. According to one article (addressing COR in tennisTennis
Tennis is a sport usually played between two players or between two teams of two players each . Each player uses a racket that is strung to strike a hollow rubber ball covered with felt over a net into the opponent's court. Tennis is an Olympic sport and is played at all levels of society at all...
racquet
Racquet
A racquet or racket is a sports implement consisting of a handled frame with an open hoop across which a network of cord is stretched tightly. It is used for striking a ball in such games as squash, tennis, racquetball, and badminton...
s), "[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
International Table Tennis Federation
The International Table Tennis Federation is the governing body for all national table tennis associations.-Founding history:The ITTF was founded in 1926, the nine founding members being Austria, Czechoslovakia, Denmark, England, Germany, Hungary, India, Sweden and Wales...
specifies that the ball must have a coefficient of restitution of 0.94 . For a hard linoleum floor with concrete underneath, a leather basketball has a COR around 0.81-0.85 .
Equation
Picture a one-dimensional collision. Velocity in an arbitrary direction is labeled "positive" and the opposite direction "negative".The coefficient of restitution is given by
for two colliding objects, where
is the final velocity of the first object after impact
is the final velocity of the second object after impact
is the initial velocity of the first object before impact
is the initial velocity of the second object before impactEven 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:
, where
is the scalar velocity of the object after impact
is the scalar velocity of the object before impactThe coefficient can also be found with:
for an object bouncing off a stationary object, such as a floor, where
is the bounce height
is the drop heightFor two- and three-dimensional collisions of rigid bodies, the velocities used are the components perpendicular to the tangent line/plane at the point of contact.
Speeds after impact
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.
- and
where
is the final velocity of the first object after impact
is the final velocity of the second object after impact
is the initial velocity of the first object before impact
is the initial velocity of the second object before impact
is the mass of the first object
is the mass of the second objectDerivation
The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). Using the notation from above where
represents the velocity before the collision and 
after, we get:
Solving the momentum conservation equation for
and the definition of the coefficient of restitution for 
yields:
Next, substitution into the first equation for
and then re-solving for 
gives:
A similar derivation yields the formula for
.