# The Physics of Collisions

Where can I find a physics book that explains how to calculate the result of collisions? For example, say I want to simulate a game of pool. How would I go about doing this?

Relevant info:

Things that collide at angles separate at 90 degree angles.
Momentum is almost totally conserved in pool.
The equation for momentum is: p = mv. p is momentum, m is mass, and v is velocity.

Any introductory textbook on physics - mechanics in particular - should have information on this.

In a collision, total momentum is always conserved. As Qwertyasdfg wrote, p=mv. So total momentum is p = m1 v1 + m2 v2, where m1 and m2 are masses of the first and second ball, etc. Note that momentum is a vector, so it’s conserved in each dimention. That is, Px = M1 V1x + M2 V2x is conserved, where V1x is the speed of the first ball in the X direction, etc. And so is Py.

Where the apostrophe denotes the quantities after the collision. But that’s not all! That doesn’t constrain the results enough. While momentum is always conserved, energy is not always conserved. If the two balls collide head-on and stick together, obviously the kinetic energy after the collision is less than before. If the collision is elastic, however, energy is conserved. It’s usually safe to approximate pool ball collisions as elastic. In this case, E = 1/2 v^2 is conserved. This is not a vector.

So now you have three equations you can use:

M1 V1x + M2 V2x = M1 V1x’ + M2 V2x’
M1 V1y + M2 V2y = M1 V1y’ + M2 V2y’
1/2 M1 V1^2 + 1/2 M2 V2^2 = 1/2 M1 V1’^2 + 1/2 M2 V2’^2

That’s all. Note that this still doesn’t constrain you to a single answer. That’s because if the collision is slightly offset, the balls fly away at different angles. But at least with these equations, if you know the velocity of each ball before the collision and angle of one of the balls after the collision, you can calculate the velocity of the balls after the collision. (“velocity” of course includes angle and speed)

Oops, the first sentence on the third paragraph should have been in the last paragraph, sorry.

Then what equation(s) do we need to add to get that answer?

Doesn’t this only apply if one ball is stationary and the other is moving?

Pool is actually a pretty simple example to use. I’d like to write an extension for a computer animation program that allows objects to behave according to physics. The first step in doing that is figuring out collisions. I know all about momentum, both linear and angular. I know about moments of inertia and centers of mass. But if the two (or more) objects in a collision can have any shape, what then? The computer could determine which points on the two objects actually collided, but how can you determine just how much momentum gets transferred? What about friction, both linear and rolling? I realize a full explanation of such things is beyond the scope of a simple post, but where can you find a book with such information?

**I know about moments of inertia and centers of mass. But if the two (or more) objects in a collision can have any shape, what then? **

Sheesh, you could have said that in the OP! It’d have saved me half an hour of typing. So you have all the initial conditions (size of balls, velocity, offset, etc.) and want to know the velocities of the balls after the collision? Then you’d have to model the actual interaction between the two balls, there’s no simple formula for that. As a first degree approximation, I’d assume that the force between the two balls are perpendicular to the surfaces at the contact point. That should constrain the solution to just one case.

I don’t know any good books off hand, sorry.

Qwertyasdfg, you’d be surprised how many pool players don’t know that colliding things separate at 90-degree angles.

The aspect about pool that I’d like to add is that the energy of the collision is nearly conserved for the brief instant after the collision, but then you have to factor in the rotation of the balls. When two balls hit, the first ball will (probably) be rolling as it hits, and will transfer its speed of translation to the second ball in a way described by freshman physics. But after the collision, the first ball will still be spinning at the same rate it was, which will no longer be the right spin for the new speed its’ going. Likewise, the second ball will start travelling in its new direction with no rolling, so it will slow down until its rolling matches its speed. This slow-down is a major factor, and explains why the energy in real-world colliding balls dissipates so rapidly. The energy is lost in the brief moments after the collisions when the balls are sliding on the cloth (even if it’s Simonis 860 like I have).

Also, how would you account for forward spin and backspin?