Hmm. You’re definitely right that my analysis (if it deserves so dignified a name) assumes all collisions are head-on, which is wrong. If “the x-axis” (i.e., the normal vector in the collision) is oriented randomly then everything still works out proportional to the kinetic energies, but as you say that might not be the case. I think this is basically the same issue as the possible bogosity of the “possibly-bogus assumption” in my original analysis.
Dealing with this all properly feels like more work than I want to do right now, though :-).
Hmm. You’re definitely right that my analysis (if it deserves so dignified a name) assumes all collisions are head-on, which is wrong. If “the x-axis” (i.e., the normal vector in the collision) is oriented randomly then everything still works out proportional to the kinetic energies, but as you say that might not be the case. I think this is basically the same issue as the possible bogosity of the “possibly-bogus assumption” in my original analysis.
Dealing with this all properly feels like more work than I want to do right now, though :-).