Physics of Billiards

Physics of Billiards

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The Physics of Billiards
Newton's Laws

First Law: An object at rest stays at rest. If it is moving, the object will continue to move with the same velocity.

Second Law: The net force on an object is equal to the product of the objects mass and its acceleration.
(F = ma)
Once the cue ball begins to roll there are no net external forces acting in the two-ball system; therefore the a must be = 0. Acceleration is the rate of change of velocity. If acceleration is 0 there is no change in velocity. When the two balls collide the only forces acting are internal and they do not affect the net force. This means that the center of mass of the system continues to move forward with the same velocity and direction after the collision.

Third Law: When two objects interact, the forces acting on them from each other are always equal in magnitude and opposite in direction.

Elastic: The Kinetic Energy of the system is conserved after the collision.
Ex. The collision of a cue ball with an object ball.
Head on: The Kinetic energy of the cue ball is transferred almost entirely to the object ball with a small amount of energy lost in sound.

The two object system is closed and isolated so linear momentum is conserved and the collision is elastic so the kinetic energy is conserved. The balls are equal in mass so:

m1v1i = m1v1f + m2v2f (linear momentum)

½ m1v1i2 = ½ m1v1f2 + ½ m2v2f2 (kinetic energy)

v1f = [(m1 - m2)/(m1 + m2)] v1i

v2f = [2m1/(m1 + m2)] v1i
If m1 = m2, the above equations reduce to v1f = 0 and v2f = v1i

Basically the cue ball is initially moving, stops suddenly when it hits the object ball at initially at rest which after the collision takes off with the initial speed of the cue ball.
After Collision

A rolling object has two types of kinetic energy.

Rotational Energy: ½ Icomw2

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Due to the rotation around its center of mass.

Translational Energy: ½ Mv2com
Due to the translation of its center of mass.
Kinetic Energy = ½ Icomw2 + ½ Mv2com

Exact point at which slide becomes rolling: Vp= R*w

The harder you hit the ball, the farther it slides before the ball is rolling (depends on the initial velcocity and spin). The faster speed makes the ball slide farther before the friction in the surface causes natural follow.
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