The Maths Of Snooker

01/09/2016 15:11 | Updated 01 September 2016

This is a short introduction to the maths of snooker. There are two basic approaches to analysing snooker mathematically: one is the scoring system and the other is the mechanics of the balls in motion.

The game consists of 22 snooker balls: one white cue ball, 15 red balls and six balls of other colours. Each non-white coloured ball corresponds to a number of points when sunk into a pocket, or "potted": 1 for red, 2 yellow, 3 green, 4 brown, 5 blue, 6 pink and 7 black. The highest score one can obtain in a single visit to the table is 147; it is the sum of the points of six red, six black and each of the coloured balls. See this for details.

The above is simple enough. Now let's examine the physics behind potting the balls.

First of all, Newton's first law states that a body remains at a constant velocity or at rest unless an external force is applied. Since snooker balls come to rest after being struck, there is indeed an external force acting on it -- the friction on the table. Yes, I mean the soft, grassy table.

Secondly, the friction is directly proportional to the normal reaction (force) on each ball, where the normal reaction is the force the table exerts perpendicularly on the ball due to the latter's weight. Here we note that physicists distinguish between mass and weight: the mass of an object is the amount of matter packed into it, while its weight is the measure of the gravitational pull on it. In other words, the weight of the ball determines the normal reaction "pushing" against it, and the normal reaction on the ball determines the friction acting against the ball. The proportionality constant in the friction-normal reaction relationship is called the coefficient of friction and is determined by experiment.

Thirdly, to know where a ball would end up, we need to find its displacement. According to the equation of motion v² - u² = 2as, where v is the final velocity (0 m/s (metres per second) for a ball at rest), u is the initial velocity (in m/s), a is the acceleration (in m/s², metres per second squared) and s is the displacement (in m, metres), we have to find its acceleration and initial velocity from its point at rest. By Newton's second law of motion, we can find the acceleration by dividing the force acting on the ball by the ball's mass. The force on the ball is the difference between the force used to strike the ball and the friction acting on it.

So far we have assumed that the cue strikes the ball at its centre, Sometimes the cue strikes the cue ball at an angle, causing the cue ball to steer away from the cue. This is because of a concept known as momentum: the product of the mass and the velocity of the ball. The greater the momentum of an object, the harder it is to get it into motion, or to slow it down if it is moving. Interestingly, momentum can be resolved into perpendicular components: one along the hitting direction of the cue and another perpendicular to the hitting direction. When a cue strikes the cue ball head-on, the perpendicular component is zero, but when it strikes the ball at an angle, the perpendicular component is nonzero and pushes the ball away from the cue.

So the next time you watch or play a game of snooker, don't forget to think about the maths behind it!