Thermal Collisions: Billiard-Ball Example | Problem 6.40

[Doom of Doom]

This is a homework question from my thermal physics class. (Problem 6.40 from Schroeder's Introduction to Thermal Physics).

Homework Statement

"You might wonder why all the molecules in a gas in thermal equilibrium don;t have exactly the same speed. After all, when two molecules collide, doesn't the faster one always lose energy and the slower one always gain energy? And if so, wouldn't repeated collisions eventually bring all the molecules to some common speed? Describe an example of a billiard-ball collision in which this is not the case: the faster ball gains energy and the slower ball loses energy. Include numbers, and be sure that your collisions conserve energy and momentum."

Homework Equations

Because the particles are all the same mass, the conservation of energy and momentum become:

$$v_1^2 + v_2^2 = v_1'^2 + v_2'^2$$
$$v_1 + v_2 = v_1' + v_2'$$

The Attempt at a Solution

I was thinking that it would be possible if two molecules were moving perpendicular to each other. If the fast one was moving along, then it was hit in the side by another (slower moving) molecule, the fast molecule would keep its velocity in the direction it was initially traveling, and gain velocity in the perpendicular direction, thus increasing its overall velocity. Is this the only case?m

 

[DrClaude]

Yes. Any collision that is not head on will lead to a "redistribution" of the velocity among the different Cartesian directions, which can lead to an increase or a decreas in speed.