Coefficient of restitution

A bouncing ball captured with a stroboscopic flash at 25 images per second. Ignoring air resistance, the square root of the ratio of the height of one bounce to that of the preceding bounce gives the coefficient of restitution for the ball/surface impact.

The coefficient of restitution (COR), or bounciness of an object is a fractional value representing the ratio of velocities after and before an impact. An object with a COR of 1 collides elastically, while an object with a COR < 1 collides inelastically. For a COR = 0, the object effectively "stops" at the surface with which it collides, not bouncing at all.

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[1][2][3]. 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 motorhome.

An important point: the COR is a property of a collision, not necessarily an object. For example, if you had five different types of objects colliding, you would have $\scriptstyle {5 \choose 2}=10$ 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 and finally rising 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 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 an arbitrary direction is labeled "positive" and the opposite direction "negative".

The coefficient of restitution is given by

$C_{R}={\frac {v_{b}-v_{a}}{u_{a}-u_{b}}}$

for two colliding objects, where

$v_{a}$ is the scalar final velocity of the first object after impact

$v_{b}$ is the scalar final velocity of the second object after impact

$u_{a}$ is the scalar initial velocity of the first object before impact

$u_{b}$ 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:

$C_{R}={\frac {-v}{u}}$, where

$v$ is the scalar velocity of the object after impact

$u$ 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 a stationary object, such as a floor, where

$h$ is the bounce height

$H$ is the drop height

For 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.

$v_{a}={\frac {m_{a}u_{a}+m_{b}u_{b}+m_{b}C_{R}(u_{b}-u_{a})}{m_{a}+m_{b}}}$

and

$v_{b}={\frac {m_{a}u_{a}+m_{b}u_{b}+m_{a}C_{R}(u_{a}-u_{b})}{m_{a}+m_{b}}}$

where

$v_{a}$ is the final velocity of the first object after impact

$v_{b}$ is the final velocity of the second object after impact

$u_{a}$ is the initial velocity of the first object before impact

$u_{b}$ is the initial velocity of the second object before impact

$m_{a}$ is the mass of the first object

$m_{b}$ is the mass of the second object

Derivation

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 $u$ represents the velocity before the collision and $v$ after, we get:

${\begin{aligned}&m_{a}u_{a}+m_{b}u_{b}=m_{a}v_{a}+m_{b}v_{b}\\&C_{R}={\frac {v_{2}-v_{1}}{u_{1}-u_{2}}}\\\end{aligned}}$

Solving the momentum conservation equation for $v_{a}$ and the definition of the coefficient of restitution for $v_{b}$ yeilds:

${\begin{aligned}&{\frac {m_{a}u_{a}+m_{b}u_{b}-m_{b}v_{b}}{m_{a}}}=v_{a}\\&v_{b}=C_{R}(u_{a}-u_{b})+v_{a}\\\end{aligned}}$

Next, substitution into the first equation for $v_{b}$ and then re-solving for $v_{a}$ gives:

${\begin{aligned}&{\frac {m_{a}u_{a}+m_{b}u_{b}-m_{b}C_{R}(u_{a}-u_{b})-m_{b}v_{a}}{m_{a}}}=v_{a}\\&\\&{\frac {m_{a}u_{a}+m_{b}u_{b}+m_{b}C_{R}(u_{b}-u_{a})}{m_{a}}}=v_{a}\left[1+{\frac {m_{b}}{m_{a}}}\right]\\&\\&{\frac {m_{a}u_{a}+m_{b}u_{b}+m_{b}C_{R}(u_{b}-u_{a})}{m_{a}+m_{b}}}=v_{a}\\\end{aligned}}$

A similar derivation yeilds the formula for $v_{b}$.

See also

• Collision
• Elastic collision
• Inelastic collision
• Resilience

References

• Cross, Rod (2006). "The bounce of a ball" (PDF). Physics Department, University of Sydney, Australia. Retrieved 2008-01-16. In this paper, the dynamics of a bouncing ball is described for several common ball types having different bounce characteristics. Results are presented for a tennis ball, a baseball, a golf ball, a superball, a steel ball bearing, a plasticene ball, and a silly putty ball. {{cite journal}}: Cite has empty unknown parameters: |month= and |coauthors= (help); Cite journal requires |journal= (help)

External links

• Wolfram Article on COR
• Bennett & Meepagala (2006). "Coefficients of Restitution". The Physics Factbook.{{cite web}}: CS1 maint: multiple names: authors list (link)
• Chris Hecker's physics introduction
• ``Getting an extra bounce" by Chelsea Wald
• FIFA Quality Concepts for Footballs - Uniform Rebound