The Academy of Quantitative Billiards

BOUNCING BILLIARD BALLS: PART 2

That branch of mathematical physics known as Mechanics has for a long time been able to predict what happens when two objects collide. However, since this area of science is seldom discussed in English Billiards/Snooker magazines, we at the Academy of Quantitative Billiards have decided to set out some of the basic ideas. Having read and digested our articles, the skilled player will be able to calculate for each shot the required speed and direction for the cue. He/she will thus become unbeatable and the governing body will be forced to buy up and destroy all copies of the BQR containing the magic formulae. That is one possible outcome, anyhow...but read on!

To discuss collisions scientifically, clear ideas about the mass and the velocity of an object are needed: it is not quite correct to say they are the same thing as weight and speed. An object in orbit round the planet is weightless, but it retains the same mass as it has when on the ground: Mass means how much matter is present in the object. Velocity is speed in a specified direction: a car approaching traffic lights at 30mph has the same speed as one approaching them at 30mph from the opposite direction, but not, because of the difference in direction, the same velocity. The scientific way of saying this is that if one car has velocity V, the other has velocity -V. The difference in velocity (the approach or relative velocity) is V - (-V) = 2V; were both cars going in the same direction, it would be V - (+V) = 0. This illustrates the importance of direction. Finally, we define the momentum of an object as the mass multiplied by the velocity.

Two objects having known mass and known initial velocity collide. What happens afterwards? Just what the objects are is not for the moment important; (cues and balls will be introduced later). Now, to allow the mathematics of the problem to be more manageable, it helps to assume that no frictional forces are present in the system, so that the speeds of all the objects only change as a result of the collision. A collision between two "coasting" hovercraft would come close to this ideal.

The diagram sets out the mathematical model for the collision. The first object involved in the collision is referred to using capital letters: M for its mass, U for its velocity before the collision and V for its final velocity. The second object is labelled with the corresponding lowercase letters m, u and v.

Fig l: The collision of two objects as the mathematician sees it.

The mathematician now writes down two equations which have been verified experimentally by the physicists (using objects not unlike mini-hovercrafts). These are statements: the first is to the effect that before and after the collision, the total momentum of the objects involved is the same and the second deals with the extent which momentum is actually transferred during the collision. These equations are then manipulated, (still using only the letters m, u..., etc.) to give new equations which give the quantity desired in terms of all the others.

Now for the billiards application, we can assume we know the masses m and M of the objects (the balls or cues) and their velocities before the collision (u and U), so we are interested in knowing their respective final velocities (v and V). In our present application, one of the objects is assumed to be at rest to start with (it will be either the cue ball in the cue-hits ball problem, or the object ball in the ball-hits-ball problem).

The only additional information required is the elasticity of the impact, which, as was discussed last article, is measured by a quantity e which ranges in value between 0 and 1. A value of 0 means that the collision is sticky, as when a bullet enters a block of wood or when a ball of putty is dropped onto a hard floor; on the other hand, a value of 1 corresponds to a perfectly elastic collision and would be realised if a ball dropped on the same floor bounced back to the same height as it was dropped from. In reality, collisions are described by a value of e between 0 and 1, so that e is a fraction.

We now look at what Mechanics predicts for the two cases of e=0 and e=l, and afterwards will be able to say that reality will lie "somewhere in between', maybe closer to one extreme than the other. We get these results:

These equations sum up quite a few things between them. As will be seen in the next article, they are relevant to disputes, old and new, about what happens when shots are made on the table.

Exploding Balls

It was reported in a recent article (BQR 8) that the earliest synthetic billiard balls were reputed on occasion to explode on contact. If true, it should be pointed out that the equations developed above are not expected to apply!

However, we at the Academy wonder if there is any truth in this legend. The material for these first composition balls was celluloid, one of the earliest plastics, which is prepared from cellulose nitrate, camphor and ethanol. The first named compound is sometimes confused with cellulose trinitrate, better known as the explosive, gun cotton. The trinitrate is prepared by a different process, however, and would be unlikely to be present in significant amounts in the finished ball. Does anyone know of a well-documented case?