The Mechanics of Snooker

1
The Mechanics of Snooker

• John James Potts
• Collingwood College

2
Introduction

• The game of snooker.
• The elasticity of snooker balls.
• The motion of snooker balls.
• The mechanics of collisions.
• How the mechanics can help a player.

3
The Game

• Two players.
• The object of the game is to score more points
  than your opponent in each frame.
• Points are accumulated by potting a red followed
  by a colour.
• A players visit to the table is ended by missing
  a pot or making a foul stroke.
• Colonel Sir Neville Chamberlain conceived the
  game in 1875.

4
Elasticity of Snooker Balls

• Elasticity is a measure of an objects ability to
  deform and subsequently regain its shape.
• Newtons Law of ImpactIf two bodies with
  initial velocities u1 and u2 collide so that they
  have final velocities v1 and v2. Then,
• where e is the coefficient of restitution for the
  two bodies, measured along the line of centres of
  the impact.
• e ? 0,1

5
Elasticity of Snooker Balls

• What is the line of centres, in the context of
  snooker ball collisions?
• The line of centres is the line which, at the
  point of contact, runs through the centre of both
  balls.

Line of Centres
6
Elasticity of Snooker Balls

• What is the value of e for snooker balls?
• Newtons Law of Impact for this system becomes
• Total Energy at A Total Energy at C before
  collision
• Total Energy at C Total Energy at B after
  collision

• Hence
• Typical value 0.4

7
Oblique Cushion Impacts

• Ball hits the cushion at an angle.

Cushion

• Assume no frictional force acting or spin.

8
Oblique Cushion Impacts

• After impact, the speed of the ball parallel to
  the cushion remains unchanged

• Using Newtons Law of Impact along line A, the
  speed of the ball after impact perpendicular to
  the cushion is

• Together, give the angle ? as the function

• Independent of the speed u.

9
Oblique Cushion Impacts

• Using the typical value of e 0.4

• Known as the slide off the cushion.

10
Friction

• Introduce friction F
• µ is the coefficient of friction between the ball
  and cloth.
• Assume it is constant.
• Typical value
• Friction has two effects

• Linear Deceleration
• Using Newtons Second Law
• We get

• Angular Acceleration
• Equation of motion
• Where I is the balls moment of inertia about
  its centre.
• Given

11
Spin

• Hitting the cue ball in various places generates
  various types of spin.
• Spin can described by angular velocity.

• Affects the motion of the cue ball.
• Changes the trajectory of the cue ball.
• Hitting the centre of the ball creates no initial
  spin a stun shot.

12
The 90o Rule

• Assume

• The collision is perfectly elastic e 1.
• The coefficient of friction between the balls is
  negligible.
• Balls have the same mass.

• Using Conservation of Momentum and Conservation
  of Energy

• This can only be true if

or

• Cue Ball trajectory post-impact is tangential to
  the point of impact.

13
Cue Ball Trajectory

• Cue ball trajectory for any cut angle, speed, and
  spin.

• Straight line motion will occur when natural roll
  commences.

• Assume
• Perfectly elastic e 1.
• The coefficient of friction between the balls is
  negligible.
• Balls have equal mass.

• Assumptions reasonable
• Typical value of e 0.94.
• Typical value of µ 0.06.

14
Cue Ball Trajectory

• Specify (x,y) coordinates.
• To formulate an expression for the trajectory,
  use linear and angular equations of motion.
• Consider backspin and topspin shots only
  rotation about thex-axis

• Where v is the initial speed and ? is the angular
  velocity about the x-axis, immediately after the
  point of impact.

15
Cue Ball Trajectory Topspin
16
Cue Ball Trajectory Backspin
17
Summary

• Governed by mechanics.
• Understanding the mechanics improves a player.
• Can be used to present simple geometry to complex
  mechanics.
• Practical approach to teaching mechanics.
• Questions?