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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.

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john mangual
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  • @KyleKanos He is incrementing time $\Delta t = 0.02s$ while I am asking to compute the collisions ahead of time. It is therefore an exercise in geometry more than anythong. Also both my spheres have the same mass - which I forgot to specify - so mass should cancel out for now. – john mangual Apr 06 '14 at 21:13
  • this is not homework but it could be... – john mangual Apr 06 '14 at 21:14
  • Curiously, both answers of the question I linked completely ignore the $\Delta t=0.02$ requirement of the OP and deal entirely with the conservation of momentum and conservation of energy. – Kyle Kanos Apr 06 '14 at 21:16
  • @KyleKanos The checked answer says Suppose that you've determined that two objects are going to collide within the next time step. I don't know the objects are going to collide yet. That is what I am asking. – john mangual Apr 06 '14 at 21:20
  • Your question is then actually answered in the linked question. – Kyle Kanos Apr 06 '14 at 21:23
  • Are the balls in free space, or are they confined? What is the area (and dimensions) of confinement? Do the balls behave the same way with the walls as with each other? BTW, anyone for a game of Pong? – LDC3 Apr 07 '14 at 03:21

2 Answers2

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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 $-mV$

DJohnM
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The criteria is

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

BAM!

John Alexiou
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