Dear Alex,
COEFFICIENT OF RESTITUION (COR)
The COR is a measure of the bounciness or rebound capacity of a ball after impact deformation. The concept of COR is the ratio of energy after impact over energy before impact, taking into account the loss of energy into heat and noise. What remains is the capacity of the object to deform at impact and rebound back to shape. In simpler terms, it is the velocity of the ball an instant before impact divided into the velocity just after impact. A ball that is hit at 100 mph and then leaves the club face at 70 mph has a COR of 0.70. The method for testing the ball COR is based upon the formula that the COR ("e") is the square root of the rebound height (h) divided by the drop height (H) onto a uniformly dense and non-deforming plane surface (an "anvil" or a concrete floor). e=SQR(h/H). Hence, a ball dropped from 10 feet that rebounds 6 feet has a COR of e= SQR(6/10), or e=0.775.
The COR of putty is 0 -- no bounce at all, and the COR of a steel ball is close to 1. A "superball" has a COR of about 0.94. If a golf ball did not have substantially less than a COR of 1, it would shatter under the impact forces in golf. Daish, in his The Physics of Ball Games (1972), recites that a good golf ball has a coefficient of restitution (COR) of about 0.70 in the context of a driver shot (pp 15-18), and that the USGA effectively limits ball COR to protect the integrity of the game on existing courses, via is distance and speed limits in testing balls. Balls of course differ. For putting, the main determinant of COR is cover hardness, with surlyn covers having a higher COR than balata, which means that identical blows with a putter send a surlyn ball farther than a balata ball. The USGA in calibrating its test equipment uses a standard ball. The COR differs with the temperature fairly substantially. This suggests that you should obtain some sort of "standard" ball and directly test its COR under constant temperature for testing and pre-testing storage. The COR also differs somewhat with the impact speed associated with drops from different heights: the higher the drop height, the lower the COR, but not by much. The reason is that the higher drop produces greater deformation and hence more energy loss. A superball COR varies from 0.945 at 0.7 meters to 0.940 at 0.9 meters. [G. Guercio & V. Zanetti, Determination of gravitational acceleration using a rubber ball, Am J Phys 55(1), Jan 1987: 59-63.] Some balls are not manufactured according to specifications, and may be "hot" balls or "cold" balls. With rubber-wound cores, a winding that is too tight produces a "hot" ball that flies farther for the same blow, and is also more likely to destruct in play. [Cochran and Stobbs, Search for the Perfect Swing (1968), p 168.] All this means that a putted ball is likely to have a COR at the high end of the range, definitely higher than that for a driver shot, and the putting COR needs to be measured in the regime of energies typical for putting.
The usual assumption for irons is that the clubs do not have enough deformation in impact with a ball to worry about the club face COR, but this is not really the case with drivers and putter faces. Today's drivers have a "trampoline effect" in the face design that somewhat absorbs the ball into the face on impact and then shoots it out, so the rebound of the driver face is a significant factor in the resulting ball speed. A similar situation results with modern inserts (urethane, ball cover material) etc., as well as with pixelated faces and softer metals. In fact, the way the putter head is attached to the shaft and perhaps even shaft flex properties can influence putter face COR. Accordingly, the COR in any given putt is a combination of the putter face and the specific ball, under temperature and speed-of-impact parameters. The USGA tests driver faces by firing a standard ball at the face and measuring the ball's in-coming and out-going velocity. The current limit is a test COR of 0.822 +/- 0.008. [See
USGA Test Procedures.]
Similarly, putter face inserts substantially alter impact duration. The idea is that the soft deforming properties of the face absorb and cradle the ball in a depressed cup-like area; that the ball rebound is reduced; but that the putter face rebound plus diminished ball rebound together make up the difference; that the prolonged "dwell time" of ball on the putter face enhances control in some way. Fisher Golf has test data on its website showing that its insert prolongs impact duration from a metal face dwell time of 1.0 ms to its KevFlex insert dwell time of 2.2 ms.
These charts show how a soft insert increases impact duration:
IMPACT DURATION / DWELL TIME
The full swing with an iron typically has an impact lasting about 0.5 milliseconds, or 5 10,000th of a second, 0.0005 seconds. In a driver shot, the driver moves about 3/4th of an inch while in contact with the ball. [Cochran and Stobbs, Search for the Perfect Swing (1968), p 144.]
The typical way to measure the duration of impact is to metalicize the ball and the impact surface, and have the impact complete a voltage circuit that displays the "contact" time onto an oscilliscope. Another way is to use high-speed photography (e.g., 8,000 frames per second) and simply count the frames of contact, carefully noting the point of initial contact and the point of separation. Daish states that impact duration varies with impact speed by a power law of F=kx^1.5 (p 102). His Fig. 10.1(a) on this page shows an impact duration of 1.5 ms at a speed of about 0.5 m/s, with the impact duration dropping to 0.5 ms at about 5 m/s, and staying at this plateau thereafter. He also provides putter impact speed data in his Fig. 11.3 (p 115) for a 15 m putt (7 m/s), a 6 m putt (4.5 m/s), and a 2 m putt (3 m/s). This would indicate that typical putter impact durations range from 1.5 ms to 0.5 ms, depending upon distance of putt (on standard green speeds). On page 150, he states that a 10 m putt requires an initial putter impact speed of 4 m/s on a green with a coefficient of friction of about 0.4. Presumably, the green referred to in Fig. 11.3 is slower, with a higher coefficient of friction.
ADDITIONAL RESOURCES:
Coefficient of Restitution
Coefficient of Restitution -- from Eric Weiss
Coefficient of Restitution - Definition
Golf Digest: Everything You Need to Know About COR
The Physics of the Golf Club
Newsday.com - Golf Physics
The physics of a golf swing
Frankly Golf
Swinging At Golf Balls And Science Lessons
Physics Laboratory 6
Coefficient of restitution and collisions
Mass of a Golf Ball
EXPERIMENT 18: COEFFICIENT OF RESTITUTION
Bouncing Ball Demonstrations
Bouncing Ball Demonstrations
golf ball aerodynamics physics
UNITED STATES GOLF ASSOCIATION testing Protocol 1999
Avant Putting Balls
Sutherland Golf - Avant - Coefficient of Restitution
ESPN.com - An additional acronym: MOI
Elastic Collision Between Different Masses
Physics - Martin Baker
Clubs and Balls
Re: A question concerning a ball?
Golf Digest: The physics of a putted ball
NASA Aeroquiz - "cold" baseballs for the visiting team
Scigolf Presents Jack Kuykendall's Myths of Golf
Golf Ball Project
World Golf: Sticks & Stones talks with former USGA technical guru
5.01.09.05 - Putter-Ball Impact
Golf Club Technology
Rice Kinesiology Department
Hireko Golf Components - Dr Leon Seltzer, What happens in that one-half millisecond?
The Science Of Choosing A Golf Ball
USGA outlines ball-testing strategy
Golf ball and golf history - History & evolution
USGA provides details of proposed new golf ball test
COEFFICIENT OF RESTITUTION
Coefficient of Restitution
Coefficient of restitution and collisions
Does the temperature of a football (or baseball) matter?
The MATH and PHYSICS of soccer! (soccer ball COR=0.58)
1N-11 Anvil and Balls
Rolling as a “continuing collision”
Force of a Golf Club on a Golf Ball
Golf Ball Velocity ISM
eGolf Weekly Golf Club Reviews--2001 Putter of the Year (Pixl S1.8)
USGA Golf House, Headquarters, and Museum - balls fired at metal block (450 millionsth of a sec in contact)
The Science of Golf Balls
Golf Putters Made with Bayer Polymers’ TPU -- STX putters with increased dwell time
Business Wire: TearDrop Golf Company Launches New TD Select Putter - decreased dwell time for better control with dimpled face -- 7 Sep 2000
Golf Digest: Haute new putters - Copper stix - soft metal for increased dwell time, Apr 2001
AllSports (tm) Putter Works of Art-- STX putter up to three times longer dwell time
FisherGolf - dwell time and ball rebound
Hustler Golf putters with tiny face particles increase dwell time
Lindsay putters -- dwell time not a source of topspin
Cheers!
Geoff Mangum
Putting Theorist and Instructor
Geoff Mangum's PuttingZone
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