In the last article but one, an account was given of the efforts of two experienced cuemen to smite a billiards ball up the table at motorway speed or thereabouts. In fact, however, not even the 30 mph urban speed limit was broken. The question naturally arises as to how this dismal result came about and how (or if) it can be improved upon. Both players were applying what they deemed to be close to the maximum muscular effort to the shot, whilst retaining some semblance to the ordinary billiards stroke. (Cue-arm apart, they kept the body as still as possible.) Why, then, was the ball so unexpectedly slow off the cue?
As this is supposed to be a scientific column, something about the physics of colliding objects is in order and this and the succeeding article will try to expound this topic without assuming prior knowledge of the subject. The physicist discusses these collisions in terms of the two concepts of elasticity and momentum.
First, what is meant by elasticity? Consider a ball dropped onto the floor. As viewed by the physicist, the collision of the ball and floor ranges from the perfectly elastic, when the ball rebounds to the height it was dropped from (and, in the absence of air resistance, rebounds to this height indefinitely), to the perfectly inelastic, when the ball does not bounce at all. To deal with intermediate cases, which are more usual, the physicist introduces a 'coefficient of restitution', which is defined as the ratio of the speeds of the ball just before and just after the impact with the floor. This ratio is a number having extreme values of 1 or 0, but typically it is a fraction and is smaller, the less elastic the collision. (This reflects the fact that, in inelastic collisions, some mechanical energy is converted into heat). This coefficient is given the symbol 'e'.
A collision between two billiard balls is characterised by a fairly high value for e, while one between a ball and a leather cue tip is rather inelastic. The physicist is able to predict what happens in such collisions in terms of this number 'e' and of the principle of momentum conservation, which will be discussed in the next article. In the meantime, anyone with access to the snooker balls (if the editor will allow mention of these in this magazine) can get a feel for the latter principle as follows:
Set up half a dozen balls against one of the cushions and touching one another. Roll a seventh ball at medium pace along the cushion, to collide with one end of the row of balls, this ball stops dead and the ball at the far end of the row simultaneously (almost) moves off at about the same speed as the impacting ball. If instead of a single impacting ball one has two or three balls moving as a unit towards the line of balls, it is found that the same number of balls detaches from the far end, with approximately the same speed. More about this phenomenon in the next article.
It has been suggested that the traditional billiard table dimensions of 6ft by 12ft, by making the table two perfect squares joined on one side, simplify the prediction of the ball's path off the cushions when a round the table shot is being mapped out. Notwithstanding the truth or otherwise of this, why is having a table of precisely two perfect squares (when the measurements are taken from the cushion nose, of course) unlikely to be helpful? What is the ideal length, if the width from one side cushion nose to the opposite one is to be 6ft exactly?