The Mechanics of Snooker Ball Collisions

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The Mechanics of Snooker Ball Collisions

• Duncan Williamson

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Snooker history

• Developed from pool.
• Conceived in a British Army Officers mess in
  India in 1875.
• Snooker spread worldwide because of the movement
  of army personnel.
• Snooker took over from Billiards as the UKs
  dominant cue sport in the 1930s.

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Snooker / Billiard balls

• What were they made from?
• Problems that arose
• John Wesley Hyatts solution celluloid.
• Modern snooker balls cast phenolic resin.
• The most important property elasticity.

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Newtons Law of impact

• If two bodies move towards each other and
  collide, the difference between their velocities
  immediately after impact is proportional to the
  difference between their velocities measured
  along the line of centres immediately before
  impact.

Where e is the coefficient of restitution
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How do we define the line of centres?

• The line of centres is the line which, at the
  point of contact, runs through the centre of both
  balls.

B
A
?
F
?
u
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Newtons Law of impact

• If two bodies move towards each other and
  collide, the difference between their velocities
  immediately after impact is proportional to the
  difference between their velocities measured
  along the line of centres immediately before
  impact.

Where e is the coefficient of restitution
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e, the coefficient of restitution

• Dependent on the material of both the objects
  which collide together.
• The more elastic the collision, the greater the
  value of e.
• A collision with associated e of 0 would be
  totally inelastic. This is, in reality,
  impossible. However, lead and concrete collisions
  have negligible associated e-values.
• The value of e is a measure of an objects
  ability to deform and subsequently regain its
  shape.

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How can we calculate e?

• Let object B be a solid, non-moving object, e.g.
  a concrete floor.
• Consider dropping a projectile under gravity from
  height H.
• Assume air resistance negligible.
• Observe the height of the rebound.
• Newtons Law of Impact reduces to

Initial velocity 0
H
Final velocity 0
h
Velocity just before impact u1 Velocity just
after impact v1 H height dropped H height
of rebound
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Calculating e
Remember
A
We can find values for v1 and u1 using the
principle of conservation of energy.
Total energy at A Total energy at B Total
energy at C
Final velocity 0
C
This gives
B
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Comparative values for e
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Oblique impacts

• i.e. collisions which do not occur head on.
• e.g. Snooker ball collision with cushion.

Overhead view
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B
u1
v1
2. Newtons Law of Impact takes effect through
the line of centres, that is parallel to the line
B.
?
?
Mass M1
A
Mass M2
3. Total momentum of the two balls is conserved
in the direction of the line of centres.
v2
?
?
u2
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Friction

• Introducing friction F...
• More realistic.

In fact, we can model friction using the
approximation Fµmg
The coefficient of friction, µ, has a constant
value greater than zero. Its value is a
function of the smoothness of the two objects
surfaces. The greater the value, the greater the
frictional force produced.
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Screw shots
Once a cue ball has been hit, it spins and skids
for some time before commencing its pure rolling
motion.
U
Table
The laws of particle dynamics used in the
analysis of oblique impacts do not predict this
behaviour. We can study this behaviour before
pure rolling re-commences
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Skidding start

• Consider a blow directed towards the centre of
  the ball, radius a NO initial rotation is
  imparted on the ball.
• Initial velocity is . Initial rotation is
  zero.
• The effect of the friction is twofold.
• (a) It produces a linear deceleration of the
  ball.
• (b) It produces a clockwise angular acceleration
  of the ball

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(a) The linear equation

• Using Newtons second law of motion
• We obtain

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(b) Angular equation
Equation of motion
I is the moment of inertia about the ball about
its axis of spin
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Two equations of motion
Linear
Angular
Proportional to is start velocity. Inversely
proportional to the coefficient of
friction. Independent of balls mass!
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How far does it skid?
Using the standard formula
We find that
Typical values in snooker are
Application to real life
This gives a skid time of approximately 0.4s
and a skid distance of 1.1m
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A video clip demonstrating the effects of
friction
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The Mechanics of Snooker Ball Collisions

• Duncan Williamson