Finding average velocity in a velocity distribution

[Woolyabyss]

Homework Statement

A gas has a hypothetical speed distribution for N gas molecules with N(v) = Cv^2 for 0 < v < V0. Find (i) an expression for C in terms of N and v0 (ii) the average speed of the particles

Homework Equations

N/A

The Attempt at a Solution

(i)
integrating N(v) with respect to v from 0 to V0 gives
N = (C*V0^3)/3 (where is N is total number of molecules)
rearranging gives C =3*N/(v^3)
(ii)
integrating N(v)*v with respect to v gives
sum(v) = (C*V0^4)/4
subbing in for C gives
sum(v) = (3N/4)*V0

sum(v)/N = Vav = (3/4)*V0

I've been having difficulty with this question for a while and can't seem to find any similar question online so I have no idea if what I did is correct. Any help would be appreciated.

 

[andrewkirk]

N(v) = Cv^2 for 0 < v < V0

What does that mean?
In the preceding sentence we were told that ##N## is the number of molecules in the system. Now all of a sudden ##N## is being used to denote something completely different.
From the fact that later on it is integrated to obtain an expected value, I presume that ##N(v)## is supposed to be a probability density function (pdf) for the random variable ##V## that is the speed of a randomly selected molecule. But if that's the case it needs to use a different letter. For instance it would be common practice to write ##F## for the cumulative distribution function so that

$$Prob(V\leq v)\equiv F(v)$$
and then
$$f(v)=\frac{dF(v)}{dv}$$
and then specify the distribution by the equation ##f(v)=Cv^2##.

But I don't know whether that is what was intended.

 

[Woolyabyss]

What does that mean?
In the preceding sentence we were told that ##N## is the number of molecules in the system. Now all of a sudden ##N## is being used to denote something completely different.
From the fact that later on it is integrated to obtain an expected value, I presume that ##N(v)## is supposed to be a probability density function (pdf) for the random variable ##V## that is the speed of a randomly selected molecule. But if that's the case it needs to use a different letter. For instance it would be common practice to write ##F## for the cumulative distribution function so that

$$Prob(V\leq v)\equiv F(v)$$
and then
$$f(v)=\frac{dF(v)}{dv}$$
and then specify the distribution by the equation ##f(v)=Cv^2##.

But I don't know whether that is what was intended.

according to my lecture notes the dimensionless product N(v) dv gives the number of molecules having speeds in the range v to v +dv.

 

[andrewkirk]

In that case, they mean by ##N(v)## the same thing as I have written above as ##f(v)##. It's bad practice to use the same symbol ##N## for two different things, but there's not much you can do about the notation your lecturer uses.
Using that interpretation, your calcs look correct. The only flaw is that in your final conclusion you have written 'sum(v)' where one would expect you have to written the average v0, since they have asked for an expression for C in terms of N and v0.