# Demonstrating the average dot-product of relative incident velocity with CM velocity of colliding pairs of ideal gas atoms is zero

In The Feynman Lectures on Physics Vol. I Ch. 39: The Kinetic Theory of Gases an argument is given in 39–4 Temperature and kinetic energy, showing that the average value of the dot-product of the relative post-collision velocity with the common center of mass velocity of pairs of colliding gas atoms is zero. Feynman then asserts that the overall average of relative velocities dotted into corresponding CM velocities is zero.

A footnote states:

This argument, which was the one used by Maxwell, involves some subtleties. Although the conclusion is correct, the result does not follow purely from the considerations of symmetry that we used before, since, by going to a reference frame moving through the gas, we may find a distorted velocity distribution. We have not found a simple proof of this result.

Is there a "standard" derivation of equation 39-19

$$\left\langle \mathbf{w}\cdot\mathbf{v}_{CM}\right\rangle =0?$$

I am particularly interested in an argument that proceeds from the model of individual collisions. That is, one that does not appeal to such abstractions as phase space, etc.