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 typically a positive real number between 0.0 and 1.0 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. The Coefficient of Restitution is equal to the Relative Speed After Collision divided by the Relative Speed Before Collision.

$\text{Coefficient of Restitution } (C_R) = \frac{\text{Relative Speed After Collision}}{\text{Relative Speed Before Collision}}$ [1]

The mathematics were developed by Sir Isaac Newton[2] in 1687.

Further details

A COR greater than one has been argued as theoretically possible. For example, some recent articles have described super-elastic collisions in which it is argued that the COR can take a value greater than one in a special case of oblique collisions.[3][4][5] These phenomena are due to the change of rebound trajectory caused by friction. It does not mean that the collisions generate kinetic energy.

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 Saint Louis University, Baguio City 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.[6]

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. 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."[7]

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.[8] For a hard linoleum floor with concrete underneath, a leather basketball has a COR around 0.81-0.85.[9]

Equation

In the case of a one-dimensional collision involving 2 objects, Object A and Object B, the coefficient of restitution is given by:

$C_R = \frac{v_b - v_a}{u_a - u_b}$, where:
$v_a$ is the final velocity of Object A after impact
$v_b$ is the final velocity of Object B after impact
$u_a$ is the initial velocity of Object A before impact
$u_b$ is the initial velocity of Object B before impact

Even though $C_R$ does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass dependent. 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 the line of impact.

For an object bouncing off a stationary target, $C_R$ is defined as the ratio of the object's speed prior to impact to that after impact:

$C_R = \frac{v}{u}$, where
$v$ is the speed of the object after impact
$u$ is the speed of the object before impact

In a case where frictional forces can be neglected and the object is dropped from rest onto a horizontal surface, this is equivalent to:

$C_R = \sqrt{\frac{h}{H}}$, where
$h$ is the bounce height
$H$ is the drop height

The coefficient of restitution can be thought of as a measure of the extent to which mechanical energy is conserved when an object bounces off a surface. In the case of an object bouncing off a stationary target, the change in gravitational potential energy, PE, during the course of the impact is essentially zero; thus, $C_R$ is a comparison between the kinetic energy, KE, of the object immediately before impact with that immediately after impact:

$C_R =\sqrt{\frac{KE_\text{(after impact)}}{KE_\text{(before impact)}}} =\sqrt{\frac{\frac{1}{2}mv^2}{\frac{1}{2}mu^2}} =\sqrt{\frac{v^2}{u^2}} =\frac{v}{u}$

In a cases where frictional forces can be neglected (nearly every student laboratory on this subject[10]) and the object is dropped from rest onto a horizontal surface, the above is equivalent to a comparison between the PE of the object at the drop height with that at the bounce height. In this case, the change in KE is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus:

$C_R =\sqrt{\frac{PE_\text{(at bounce height)}}{PE_\text{(at drop height)}}} =\sqrt{\frac{mgh}{mgH}} =\sqrt{\frac{h}{H}}$

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 $v_b$.

References

1. ^ McGinnis, Peter M. (2005). Biomechanics of sport and exercise Biomechanics of sport and exercise (2nd ed.). Champaign, IL [u.a.]: Human Kinetics. p. 85. ISBN 9780736051019.
2. ^ "'A' level Revision:Newton's Law of Restitution". Retrieved 12 March 2013.
3. ^ Louge, Michel; Adams, Michael (2002). "Anomalous behavior of normal kinematic restitution in the oblique impacts of a hard sphere on an elastoplastic plate". Physical Review E 65 (2). Bibcode:2002PhRvE..65b1303L. doi:10.1103/PhysRevE.65.021303.
4. ^ Kuninaka, Hiroto; Hayakawa, Hisao (2004). "Anomalous Behavior of the Coefficient of Normal Restitution in Oblique Impact". Physical Review Letters 93 (15). arXiv:cond-mat/0310058. Bibcode:2004PhRvL..93o4301K. doi:10.1103/PhysRevLett.93.154301.
5. ^ Calsamiglia, J.; Kennedy, S. W.; Chatterjee, A.; Ruina, A.; Jenkins, J. T. (1999). "Anomalous Frictional Behavior in Collisions of Thin Disks". Journal of Applied Mechanics 66 (1): 146. Bibcode:1999JAM....66..146C. doi:10.1115/1.2789141.
6. ^
7. ^
8. ^ "ITTF Technical Leaflet T3: The Ball" (PDF). ITTF. December 2009. p. 4. Retrieved 28 July 2010.
9. ^ "UT Arlington Physicists Question New Synthetic NBA Basketball". Retrieved May 8, 2011.
10. ^
• 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.