The last days I watched some games of the snooker world championship at The Crucible Theatre in Sheffield. I was amazed at the incredible quality of the play. Then I started to wonder : what calculations are necessary to come to such magnificent shots ?
I could google surprisingly little on the physics of snooker. And what I found was either poor physics or poor snooker.
So I decided to give a brief overview of the physics of snooker, as I see it. By the way, my snooker capacities are very limited :-), but I love the game, and my physics knowledge limits to some practical insights, so I leave out all the theory.
To shorten this post and to reduce the complexity I will limit this overview to one single shot.
What do we want ? The object ball has to go in the pocket, the cue ball must come to rest on a very good spot to pot the next object ball.
What are our limits ? i) We can only play the cue ball. Everything that we want the object ball to do is caused by the collision with the cue ball. ii) with the cue stick we can give the cue ball a forward motion and a spin (follow, draw, side, where follow and draw correspond with top spin and back spin respectively, and of cause combinations between follow or draw and side). As for the object ball, only a forward motion is practically possible. Given the fact that there is very little friction when two balls collide the limited object ball spin is of no practical value.
( See drawing at onekaraoke.com)
OK, how will we attain these two objectives ?
First objective : the object ball must go in the pocket. The direction of the object ball is totally determined by the point where it is touched by the cue ball. When the two balls collide, simply draw a line through the centers of the two balls. This line shows the direction of the object ball. It should point straight to the center of the pocket.
This is the simplest of the two, but it is far from simple ! The direction of the object ball is limited between 180° (goes in the same direction as the cue ball, when this direction goes through the center of the object ball) and a theoretical 90° when the cue ball “kisses” the object ball extremely thinly at its side.
Second objective : the cueball must stop at the exact (approximately is mostly good enough) spot of the table where we want it to stop. Remember the cueball has two moving properties : direction and spin, so this is a lot more complicated than simply potting the object ball.
– Right after (first nanoseconds ! ) the collision : the cue ball direction is entirely determined by the touching points. This direction can be anything between 90° right to 270°(left), except for the circle segment behind the object ball. The size of this segment is determined by the original distance between the two balls.
At the moment of the impact, the impulse does not change. With a completely central impact, this means that the object ball will follow the direction of the cue ball at the speed of the cue ball, and the cue ball will remain motionless at the exact spot where it touched the object ball. The other extreme is that the cue ball misses the object ball, no need to tell what happens.
Then you have everything in between : touching very slightly the object ball will cause
i) a very little deviation and apparently nearly no loss of speed of the cue ball and
ii) a very slow movement of the object ball in an angle of just above 90°. So here is not much the player can play with to get his cue ball on the right spot after the shot.
– Later on (following nanoseconds, milliseconds, seconds …) after the collison there is something totally different in play : the spin of the cue ball. Until the cue ball hits a cushion the side spin does not play an important role (and by the way, this would be too difficult to elaborate). The vertical spin though has an huge effect on the cue ball direction.
Let us start with a head kick, straight at the center of the object ball.
There are three possibilities :
1) the cue ball has absolutely no spin when it touches the object ball. In that case the cue ball immediately stops and stays where it hits the object ball.
2) the cue ball is kicked with a draw (back spin) : after the collision the backspin and the friction with the table forces the cue ball to come back in the direction where it came from. the speed and distance is determined by the amount of spin.
3) the cue ball is kicked with a follow (top spin) : after the collision the topspin and the friction with the table forces the cue ball to follow the same direction it had before the collision. The speed and distance is determined by the amount of spin. (Note that this can cause the cue ball to disappear in the pocket too !)
It is important to realise that in 2) and 3) there is an acceleration of the cue ball (remember : in the first nanosecond after the collision it stopped every movement) after the collision, until the spin has slowed down until there is no more friction and the cue ball simply rolls further, meaning that the spinning velocity equals the horizontal velocity.
It is this acceleration phase that is interesting in the other case, where the cue ball hits the object ball at an angle : in that case we have
i) a straight movement of the cue ball at an angle to its original direction and at the same time
ii) an acceleration in either the same or the opposite direction of the original movement. This causes the cue ball to follow a curved trajectory, until the complete rolling phase. With this draw and follow spin, an excellent player is able to force the cue ball to follow nearly any direction between 0 and 180° after the collision.
And at last comes the side spin into play : when the cue ball hits the cushion. Normally it leaves the cushion at the same angle of its arrival, but at the other side of the perpendicular. With the side spin the player can increase or decrease this angle. If the angle is close to the perpendicular, it is even possible to leave at the same side as the arrival.
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