|
|
|
|
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 "stopping" at the surface with which it collides, not bouncing at all.
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.
Discussion
Ask a question about 'Coefficient of restitution' Start a new discussion about 'Coefficient of restitution' Answer questions from other users |
Encyclopedia
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 "stopping" at the surface with which it collides, 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
Picture a one-dimensional collision. Velocity in one direction is labeled "positive" and the opposite direction "negative".
The coefficient of restitution is given by:
for two colliding objects, where
is the scalar final velocity of the first object after impact
is the scalar final velocity of the second object after impact
is the scalar initial velocity of the first object before impact
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:
, where
is the scalar velocity of the object after impact
is the scalar velocity of the object before impact
The 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 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 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 (e) 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 value of e 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/sec and finally rising again as the collision speed approaches zero. This effect was observed in slow speed collisions involving a number of different metals.
The paper, published in August 1955, IMPACT STUDIES ON PURE METALS, is a Physics Master's Degree Thesis by John W. Burgeson. A copy of this paper is on the web at www.burgy.50megs.com/thesis,htm and also on WilkiCommons at .
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.
- 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 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):
See also
External links
|
| |
|
|
