To begin with, I needed to dig out Halliday & Resnick, and review the equations for collisions. There is a newer edition but I used my old book, which I re-purchased some years ago from Abebooks. I want to show the derivations of the equations for collisions between two particles in one and two dimensions. Here is a graphic to motivate us:

In the one dimensional case, of course, we do not have the angles θ and &phi. We just have masses M and m, and velocities before and after collision. These are usually designated v

_{1i}, etc. but I am going to use non-standard notation to reduce my confusion, and as a bonus, simplify the typing. We will label the velocities of the masses before collision as a (for M) and b (for m), while afterward they will be c (for M) and d (for m).

We use two of the great conservation principles of physics, for momentum and energy.

In one dimension (inelastic collision), we have:

Halliday & Resnick:

This tells us that in an elastic one-dimensional collision, the relative velocity of approach before collision is equal to the relative velocity of separation after collision.

We could also solve for d, but I will just assert that the following is true by symmetry:

(There is nothing special about m and M, so if we switch them as well as a and b, it should be OK).

Special cases of interest.

If the masses are equal, M = m and:

The particles simply exchange velocities.

Another case is where M (say) is initially at rest, then a = 0 and

Now, for equal masses, then d = 0 and M simply acquires the velocity of m, while m stops abruptly.

However, if M is at rest

*and*the masses are greatly unequal, (M >> m), then the velocity of M after the collision is still c ≈ 0, and d = -b, the small particle reverses its velocity. It just bounces off.

I'll do the 2D case in another post, and then move on to the simulation.

## No comments:

Post a Comment