Collision of two balls of finite radius

Question

This is your typical elastic collision problem except the balls have finite radius. To be clear:

• two billiard balls in the plane each with radius $\sigma$
• move at constant velocities $v_1, v_2$
• starting at positions $x_1, x_2$

What is the criterion for there to be any collision at all? When does it happen? And at what angle to they collide?

I need it to write a computer simulation of the hard sphere model.

Answer 1

I think you have too many parameters, and not all of the necessary ones

To simplify your thinking:

Change to a frame of reference in which one billiard ball is initially at rest at the origin, and the second is moving at velocity $V$ from right to left along the straight line $$y=k, \,k>=0$$

A collision will take place if and only if $k<2\sigma$.

The collision occurs when the moving ball is at a distance $2\sigma$ from the origin.

The collision will take place at a point on the stationary ball an angle $\theta$ measured counter-clockwise from the positive x-axis, where:$$\sin\theta=\frac{k}{2\sigma}$$

The collision will take place in the first quadrant; total momentum in the $y$ direction will be conserved at $0$; and total momentum in the $x$ direction will be conserved at $-mV$

Answer 2

The criteria is

$$ \mbox{center to center distance} \le \mbox{radius 1} + \mbox{radius 2} $$

BAM!