Via physics and physicists I see a story about how golf can be hazardous to your hearing. And the story botches the physics. (I don’t know if it’s the journalist or from the original journal article)
The coefficient of restitution (Cor) of a golf club is a measure of the efficiency of energy transfer between the golf club head and the golf ball. The upper Cor limit for a golf club in competition is 0.83, which means that a golf club head striking a golf ball at 100km per hour will cause the ball to travel at 83km/h.
Well, that’s just wrong. The Cor tells you about the kinetic energy, so it won’t be the same for the speed, because KE depends on v2. i.e. if a ball is dropped from 1 meter and bounces, returning to 0.83m, the impact speed is ~4.4 m/s and the return speed is ~4.0 m/s, which is 0.91 of the speed.
Another problem is that the mass of the clubhead is not the same as the mass of the ball. Even if the Cor applied to speed, the statement is incorrect. In the limiting case of Cor=1 and the ball’s mass being negligible, the ball would leave at twice the clubhead speed.
The actual equation is v = u*(1+e)/(1+m/M)
v is the ball’s speed, u is the clubhead speed, e is the Cor, m is the ball’s mass and M is the clubhead mass. (This is trivially derived using conservation of momentum and balancing the kinetic energy equation to account for the loss) Using e = 0.83, and assiming the M=4m, we see that v = 1.46u
Update — This is using a definition of Cor on terms of energy. I couldn’t find how the USGA was defining it when I was composing the post, but further research (and noted in the comments) indicates that it is indeed the fraction of the speed retained after the collision. That changes the details of the analysis, but the article’s numbers are still wrong — the ball’s speed is larger than the clubhead speed. I still haven’t found a mathematical definition of how the USGA applies this to a golf club
That makes e in the equation the square of the Cor (so e = 0.689), which means that the ball leaves the clubhead at v = 1.35u
Coefficient of restitution is actually defined as the ratio of speeds. That’s the *relative* speed prior to impact vs. the relative speed after impact. But as you say, it’s still wrong. If you assume the club/ball mass ratio >> 1, the ball should end up with a speed of 183 km/h, not 83 km/h.