"Show that particles hitting a plane boundary have travelled a distance 2λ/3 perpendicular to the plane since their last collision, on average."
(Root mean path squared) <x> = 2^(.5)λ
λ = ( 2^(.5) * n * sigma )^(-1)
The Attempt at a Solution
I already knew I needed <x cos(theta)>.
The book tells me that I can split x and cos(theta) apart by this rule:
My issue is that I do not understand how to calculate the average angle. I know the boundaries are 0 < theta < pi/2.
I would expect the average angle to be pi/4.
(NOTE: I measured the angle from the plane and should have shown measurement from the z axis down to satisfy the cos(theta).
So I figured that if I can just determine the average angle, I could plug it into <x> * cos(theta) and I would be done. However, that does not seem to work.
Upon looking at the answer, I am clueless of what they are doing in the angle integral:
I am also unsure why they are taking the mean probability distributions
And lastly, I don't understand why they included the velocity function, even though it just cancels out anyways.