A molecule has a velocity v and speed v. I've worked out (and understand) that the number of molecules in a gas with speeds between v and v+dv and moving at angles between Ө and Ө+dӨ to any chosen axis is: (1/2)nf(v)dvsin(Ө)dӨ The internet verifies this. f(v) is the speed distribution (often Maxwellian but this question is general).
Here's the question. If u is a cartesian component of v (eg v_x), show by integration that:
<u> = 0,
<mod(u)> = (1/2)<mod(v)>
The question hints that <v> is the integral from 0 to infinity of vf(v)dv divided by the same integral of f(v)dv and think I understand why. But I'm not sure how to put it in terms of u?
The Attempt at a Solution
Intuitively <u> = 0 because there are the same number of molecules going "left" as there are going "right". But I can't show it rigorously using the equation above. I've found a couple of cases where f(v) is the maxwellian distribution but this question specifically asks you to solve it generally. I keep getting myself in a muddle with the infinite limits as well.
The question is attached as a picture. I've taken photos as well of some jottings. Thanks in advance to anybody even just for some hints.
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