1. The problem statement, all variables and given/known data "Show that particles hitting a plane boundary have travelled a distance 2λ/3 perpendicular to the plane since their last collision, on average." 2. Relevant equations (Root mean path squared) <x> = 2^(.5)λ λ = ( 2^(.5) * n * sigma )^(-1) ANSWER: 3. 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: . Why is sin(theta) here? I am also unsure why they are taking the mean probability distributions and dividing them by the integral of the probability functions . And lastly, I don't understand why they included the velocity function, even though it just cancels out anyways. Thanks.