Proving Velocity Averaging: <u> = 0, <u^2> = 1/3 <v^2> & <|u|> = 1/2 <v>

[bon]

Homework Statement

The molecules in a gas travel with different velocities. A particular molecule will have velocity v and speed v=|v| and will move at an angle X to some chosen fixed axis. The number of molecules in a gas with speeds between v and v + dv and moving at angles between X and X+dX to any chosen axis is given by

1/2 n f(v)dv sinX dX

Where n is the numberof molecules per unit volume and f(v) is some function of v only.

Show by integration that:

(a) <u> = 0
(b) <u^2> = 1/3 <v^2>
(c) <|u|> = 1/2 <v>

where u is anyone cartesian component of v i.e v_x, v_y or v_z

The question says that we can take u as the z-component of v without loss of generality. Why is this? Could we equally well have taken it as the x or y-component. It then says express u in terms of v and X and average over v and X.

Homework Equations

The Attempt at a Solution

Not sure how I am meant to do this, so any help would be great - thanks!

 

[bon]

hello people??