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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).[1]

Contents

 [hide] 

1 Further details o 1.1 Sports equipment

2 Equation 3 Speeds after impact

o 3.1 Derivation 4 See also 5 References 6 External links

[edit] 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.[2][3][4] 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.[5]

[edit] 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.[6] 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].

[edit] 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 impactis the final velocity of the second object after impactis the initial velocity of the first object before impactis 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.

For an object bouncing off a stationary object, such as a floor:

, whereis the scalar velocity of the object after impactis 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 heightis 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.

[edit] 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 impactis the final velocity of the second object after impactis the initial velocity of the first object before impactis the initial velocity of the second object before impactis the mass of the first objectis the mass of the second object

[edit] 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 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 .

http://en.wikipedia.org/wiki/Coefficient_of_restitution

Definition: "Coefficient of Restitution" (often just "COR") is a technical term describing the energy transference between two objects. The coefficient of restitution of Object A is a measurement of Object A's ability to transfer energy to Object B when A and B collide. Golf club designer Tom Wishon, in our FAQ, What is COR?, provides the technical definition this way:"Coefficient of restitution is a measurement of the energy loss or retention when two objects collide. The COR measurement is always expressed as a number between 0.000 (meaning all energy is lost in the collision) and 1.000 (which means a perfect, elastic collision in which all energy is transferred from one object to the other)."

How does coefficient of restitution enter into golf? Golf clubs collide with golf balls, and the more energy that is transferred from the club to the ball in that collision, the farther the ball can travel.

Coefficient of restitution can be measured for all golf clubs, but is most commonly associated with drivers. Currently, the USGA and R&A place a limit of .830 on COR; COR cannot go above that in a club.

The term came into the popular golf lexicon as ultra-thin-faced drivers began to proliferate in the early 2000s. An effect of the thin faces is known as the "spring-like effect" or "trampoline effect": The face of the driver depresses as the ball is struck, then rebounds - providing a little extra oomph to the shot. A driver that exhibits this property will have a very high COR.

Coefficient of restitution is rarely advertised by golf manufacturers these days because almost all drivers are at the limit set by the governing bodies. Drivers above the limit are non-conforming under the rules.

http://golf.about.com/cs/golfterms/g/bldef_cor.htm

coefficient of restitution

Definition/Summary

For a collision between two objects, the coefficient of restitution is the ratio of the relative speed after to the relative speed before the collision.

The coefficient of restitution is a number between 0 (perfectly inelastic collision) and 1 (elastic collision) inclusive.

Equations

The coefficient of restitution is

C.O.R.=|v2f ⃗ −v1f ⃗ ||v2i ⃗ −v1i ⃗ |where v1i ⃗  and v1f ⃗  are the initial and final velocities, respectively, of object #1. A similar definition holds for the velocities of object #2.

While this is a useful definition for studying collisions of particles in physics, there is an alternative used to define the C.O.R. of everyday objects. In this definition, the velocities are replaced with the components perpendicular to the plane or line of impact. In the case of a 1-d collision, the two definitions are equivalent.

Be sure you know which definition of C.O.R. is the accepted practice in a given situation. For the remainder of this discussion, we use the definition in the equation shown above.

For an object colliding with a fixed object or

surface, v2i and v2f are zero, and the C.O.R reduces to:

C.O.R.=|v1f ⃗ ||v1i ⃗ |In the center-of-mass reference frame of two

  Breakdown

Physics> Classical Mechanics>> Newtonian Dynamics

See Also

Images

objects of mass m1 and m2 -- and only in that frame -- the initial and final total kinetic energies are related to the C.O.R. by

KEfKEi=C.O.R2

where

KEi = 12m1v21i + 12m2v22i

and

KEf = 12m1v21f + 12m2v22f

Scientists

Recent forum threads on coefficient of restitution

Extended explanation

Elastic and perfectly inelastic collisions

The coefficient of restitution describes the inelasticity of collisions. If the C.O.R. is 1, the collision is elastic and kinetic energy is conserved. A C.O.R of zero represents a perfectly inelastic collision; after the collision the objects stick together and, in the center-of-mass frame, have zero velocity.

Simplifying 1-d collision problems

In a 1-dimensional elastic collision (C.O.R. = 1), the conservation-of-energy equation may be replaced with

v2f−v1f=v1i−v2i

In other words, the relative velocity of the two particles has the same magnitude, but is reversed in direction, before and after the collision. By using this equation instead of the conservation-of-energy equation directly, the work of solving a collision problem is simplified as there are no squared velocity terms to deal with.

Commentary

Redbelly98 @ 06:46 PM Mar24-11

Hopefully going over to MathJax in the near future will fix the LaTeX problems.

tiny-tim @ 03:53 PM Mar24-11

Fixed missing LaTeX again. No other changes.

tiny-tim @ 12:26 PM Nov25-10

Fixed missing LaTeX. No other changes.

tiny-tim @ 02:55 AM Jul13-09

"For an object bouncing off a stationary object …" is wrong: it must be stationary both before and after, so it has to be off a fixed surface, but even then the motion must be perpendicular to the surface … explaining this really requires a comment on the direction of the constraining force, and the difference between constrained and unconstrained collisions.

Redbelly98 @ 10:21 AM Jun23-09

(EDIT: This comment is no longer relevant, since the equation has since been corrected.)

In the equation for C.O.R., should that be calculated in the center-of-mass frame? That's the only frame in which

|| Σvf || = 0 for a completely inelastic collision.

http://www.physicsforums.com/library.php?do=view_item&itemid=270

he Coefficient of Restitution

The Coefficient of Restitution (e) is a variable number with no units, with limits from zero to one.

0 ≤ e ≤ 1

'e' is a consequence of Newton's Experimental Law of Impact, which describes how the speed of separation of two impacting bodies compares with their speed of approach.

note: the speeds are relative speeds

If we consider the speed of individual masses before and after collision, we obtain another useful equation:

uA = initial speed of mass A

uB = initial speed of mass B

vA = final speed of mass A

vB = final speed of mass B

relative initial speed of mass A to mass B = uB - uA

relative final speed of mass A to mass B = vB - vA

note: in this equation the absolute of uB - uA and vB - vA are used ( |absolute| no net negative result )

 

Example

A 5 kg mass moving at 6 ms-1 makes a head-on collision with a 4 kg mass travelling at 3 ms-1 .Assuming that there are no external forces acting on the system, what are the velocities of the two masses after impact?(assume coefficient of restitution e = 0.5 )

uA = initial speed of 5 kg mass (mass A)= 6 ms-1

uB = initial speed of 4 kg mass (mass B)= 3 ms-1

mA = 5 kg       mB = 4 kg

vA = final speed of mass A        vB = final speed of mass B

 

momentum before the collision equals momentum after

hence,        mA uA + mB uB = mA vA + mB vB

also

substituting for e, mA , uA , mB , uB

we obtain two simultaneous equations

from the conservation of momentum,

5 x 0.6 + 4(-3) = 5 vA + 4 vB

3 - 12 = 5 vA + 4 vB

-9 = 5 vA + 4 vB

5 vA + 4 vB = -9        (i

from the coefficient of restitution expression,

0.5(uA - uB) = vB - vA

(0.5 x 6) - (0.5(-3)) = vB - vA

3 + 1.5 = vB - vA

4.5 = vB - vA

vB - vA = 4.5        (ii

multiplying (ii by 5 and adding

5 vA + 4 vB = -9

- 5 vA + 5 vB = 22.5

9 vB = 13.5

vB = 1.5 ms-1

from (ii

1.5 - vA = 4.5

vA = 1.5 - 4.5 = -3

vA = -3 ms-1

Ans. The velocities of the 5 kg and 4 kg masses are -3 ms-1and 1.5 ms-1, respectively.

 

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Oblique Collisions

For two masses colliding along a line, Newton's Experimental law is true for component speeds. That is, the law is applied twice: to each pair of component speeds acting in a particular direction.

Example

A particle of mass m impacts a smooth wall at 4u ms-1 at an angle of 30 deg. to the vertical. The particle rebounds with a speed ku at 90 deg. to the original direction and in the same plane as the impact trajectory.

What is:i) the value of the constant 'k' ?ii) the coefficient of restitution between the wall and the particle?iii) the magnitude of the impulse of the wall on the particle

i) There is no momentum change parallel to the wall.

 

ii) The coefficient of restitution'e' is the ratio of the speed of separation to the speed of approach:

 

iii) The impulse is the change of momentum.

Since the vertical unit vectors are unchanged, the momentum change just concerns the horizontal vector components.

hence,

 

http://www.a-levelphysicstutor.com/m-momimp-coeffrest.php

COEFFICIENT OF RESTITUTION

Well, "restitution" is what you give back. And "coefficient" means a single number that describes how much. Got it?

The coefficient of restitution is a number which indicates how much kinetic energy (energy of motion) remains after a collision of two objects.

If the coefficient is high (very close to 1.00) it means that very little kinetic energy was lost during the collision. If the coefficient is low (close to zero) it suggests that a large fraction of the kinetic energy was converted into heat or was otherwise absorbed through deformation. Let's take a closer look:

When a moving object (say a rubber ball) collides with an immobile flat surface (say a massive marble floor), the object will rebound with some fraction of its original energy. If the collision is perfectly

elastic, then the ball will rebound with all of the energy it arrived with and its rebound velocity will be the same as its approach velocity. In this case, the coefficient of restitution is said to be precisely 1.00. On the other hand, if there is considerable permanent deformation of either the object or the surface (or both) then the object will rebound with much less energy than it originally arrived with. In this case, the coefficient of restitution will be close to zero.

In the latter case, the surface was probably soft or otherwise absorptive. In the first case, the surface might have been particularly hard... or it may have been particularly springy.

And this is where the coefficient of restitution becomes important in golf.

We won't bore you with explanations of momentum, but the example above would have also worked with an initially motionless object (say, a teed up golf ball) being struck by a moving surface (the face of a driver).

There is nothing magic about this number called the coefficient of restitution. It doesn't make one driver any better or worse than another any more than a speedometer can make its car go faster or slower than another. The speedometer merely tells you how fast the particular car is going. And the coefficient of restitution merely tells you how much of the original kinetic energy remains after a collision of the clubhead with the golf ball.

The higher the coefficient of restitution, the faster the ball will be propelled by the clubhead for a given impact speed.

So why all the fuss over this figure?

It's because the USGA is trying its best to limit the influence of technology in golf. Theory being that golf should challenge the skill and savvy of the individual rather than the technology in his or her bag. And this is probably a good thing for the sport. After all, it would be very effective to go bass fishing by dropping depth charges over the side of your boat. Just scoop 'em up when they float to the surface. But would that really be bass fishing?

So the USGA has adopted limits of 0.830 on the coefficient of restitution a given clubface may have as part of its effort to define the boundaries of technology in the game.

For a discussion of the politics involved and the ultimate compromise resolution, refer to COR.

http://www.leaderboard.com/GLOSSARY_COEFFICIENTOFRESTITUTION