What determines the angle of the cushion on a pool table?

If you look at the cushions (bumpers) on a pool table, you'll see that they're not vertical. They're tilted inwards. About 10 years ago, I came across a physics exam in which one of the problems stated a set of physical assumptions and then asked the examinee to determine the optimal angle so that if a ball was rolling without slipping when it hit the cushion, it would rebound such that it was again rolling without slipping. The implication seemed to be that this was the angle used on actual pool tables.

I didn't have access to the examiner's own solution, and when I analyzed the problem myself, I found that I couldn't reproduce the actual angles found on pool tables. There seems to have been some speculation that somewhere back in the mists of time, somebody set the angle empirically by maximizing the rebound distance for balls incident along the normal, and that this angle was well explained by this type of calculation. In fact, high-speed video (Mathaven 2009) shows that in snooker, a ball that hits a cushion along the normal doesn't rebound by rolling without slipping; it slips for about 0.1 s before the torque due to kinetic friction brings it into the non-slipping state. (I don't think the physical situation is vastly different between pool and snooker.)

Some data. Let $r$ be the radius of a pool ball, and let the angle of the cushion be such that the point of contact between the ball and the cushion is above the center of the ball by a height $b$. Real pool tables have $b/r\approx 0.26$. In snooker, the coefficient of kinetic friction, as measured by Mathaven, is $\mu_k \approx .18-.24$, and the deceleration due to rolling resistance is about $a/g\approx 0.0127-0.0129$.

My question: Is there any physical explanation for the angle of the cushions on a pool table? It doesn't seem to be optimal for normal incidence, but is it possible that it's optimized in the sense of an average over all possible angles of incidence? Are there models and possibly computer simulations that are sufficiently sophisticated to address this kind of question?

Mathaven, S., et al., "Application of high-speed imaging to determine the dynamics of billiards," American Journal of Physics Vol. 77, No. 9, pp. 788-794, 2009. http://billiards.colostate.edu/physics/ajp_09_hsv_article.pdf

-
Fascinating. Another possible goal would be to retain the angle of reflection equal to the angle of incidence (on the assumption of rolling straight ahead?) despite the inelastic nature of the collision. –  dmckee Apr 29 at 14:55
One factor that was likely considered was to minimize the likely hood of high speed balls bouncing off the pool table. Perhaps the current cushion angle was found to be optimal for this situation. –  Michael Luciuk Apr 29 at 19:15
This analysis claims the 'sweet spot' of a pool ball is (7/5)r . I would have guessed that height was the reason for table design, rather than a particular angle. –  Mark Rovetta Apr 29 at 19:21
@MarkRovetta: The angle $\theta$ of the cushion is related to the height at which it touches the ball. The result you quote would be, in my notation, $b/r=0.4$, or $\theta=\sin^{-1}(b/r)=41$ deg. As noted in the question, actual pool tables have $b/r=0.26$, or $\theta=15$ deg. Their result is derived under the assumption that the force acting on the ball is horizontal, but that's not correct for a ball hitting a cushion, as discussed in the link to my own analysis. So the result of the analysis you linked to (a) doesn't match reality, and (b) is based on assumptions that don't apply here. –  Ben Crowell Apr 29 at 22:23
@BenCrowell: According to this site, the 'sweet spot' of $b/r=0.4$ isn't used because the downward push of the rebounding ball onto the surface would cause too much wear. So instead a slightly lower height is used. According to BCA specifications, billiard balls have a diameter of 2 1/4", while the cushion height is usually taken as 1 7/16" (but there is variation between manufacturers). For these values, $b/r\approx 0.28$. –  Pulsar May 22 at 7:46
show 1 more comment