Coefficient of restitution

A bouncing basketball 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) of two colliding objects is a fractional 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, while objects with COR < 1 collide inelastically. 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). The mathematics was developed by Sir Isaac Newton in 1687.

Further details

A COR greater than one is theoretically possible, representing a collision that generates kinetic energy. For example, 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.

The COR is a property of a collision, not a single object. If a given object collides with two different objects, each collision would have its own COR.

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 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 the flexing and subsequent release of stored energy, imparting greater impulse to the ball. The USGA (America's governing golfing body) has started testing drivers for COR and has placed the upper limit at 0.83. Golf balls themselves typically have a COR of about 0.78, but this varies based on the "compression rating" (a measure of the hardness of the ball). 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 shall bounce up 24–26 cm when dropped from a height of 30.5 cm on to a standard steel block thereby having a COR of 0.89 to 0.92. 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

$C_R = \frac{v_b - v_a}{u_a - u_b}$

for two colliding objects, 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

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. It is one dimensional unitless parameter defined only along line of impact.

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

Coefficient of restitution is defined as negative of ratio of relative velocity of separation and relative velocity of approach along line of impact. It is never negative(could be negative for irregular shapes). 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

This is because energy is conserved, $KE = {\frac{1}{2}}mv^2$, and so:

$C_R =\frac{v}{u}=\sqrt{\frac{v^2}{u^2}} =\sqrt{\frac{KE_\text{after}}{KE_\text{before}}}$

When the ball hits the floor, its gravitational potential energy (=mgh) is at a minimum, as h, height, cannot be any lower (i.e. the ball can't go through the floor), so all its energy is Kinetic. When kinetic energy=0 (i.e. at the peak of its motion when it momentarily stops), all the energy is potential. Thus the ratio of heights (maximum potential energy) is also a square-root ratio one:

$C_R =\sqrt{\frac{mg h_\text{after}}{mgh_\text{before}}}$

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 i.e. along line of impact.

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:

$\begin{align} & m_a u_a + m_b u_b = m_a v_a + m_b v_b \\ & C_R = \frac{v_b - v_a}{u_a - u_b} \\ \end{align}$

Solving the momentum conservation equation for $v_a$ and the definition of the coefficient of restitution for $v_b$ yields:

$\begin{align} & \frac{m_a u_a + m_b u_b - m_b v_b}{m_a} = v_a \\ & v_b = C_R(u_a - u_b) + v_a \\ \end{align}$

Next, substitution into the first equation for $v_b$ and then re-solving for $v_a$ gives:

$\begin{align} & \frac{m_a u_a + m_b u_b - m_b C_R(u_a - u_b) - m_b v_a}{m_a} = v_a \\ & \\ & \frac{m_a u_a + m_b u_b + m_b C_R(u_b - u_a)}{m_a} = v_a \left[ 1 + \frac{m_b}{m_a} \right] \\ & \\ & \frac{m_a u_a + m_b u_b + m_b C_R(u_b - u_a)}{m_a + m_b} = v_a \\ \end{align}$

A similar derivation yields the formula for $v_b$.

See also

• Collision
• Resilience

References

• Cross, Rod (2006). The bounce of a ball. 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." 
• Walker, Jearl (2011). Fundamentals Of Physics (9th ed.). David Halliday, Robert Resnick, Jearl Walker. ISBN 978-0-470-56473-8. 

External links

• Wolfram Article on COR
• Bennett & Meepagala (2006). "Coefficients of Restitution". The Physics Factbook. 
• Chris Hecker's physics introduction
• "Getting an extra bounce" by Chelsea Wald
• FIFA Quality Concepts for Footballs – Uniform Rebound
• Bowley, Roger (2009). "Coefficient of Restitution". Sixty Symbols. Brady Haran for the University of Nottingham.