1. Dimensions

Hello everybody, I am having trouble with this question:

Two common models for the resistive force experienced by a projectile in flight due to air resistance are:-

(a) Force proportional to speed so that $\displaystyle F = -k_{1}v$;
(b) Force proportional to the square of speed so that $\displaystyle F = -k_{2}v^{2}$.

Investigate the dependence of $\displaystyle k_1$ on $\displaystyle \mu$, the viscosity of air and $\displaystyle r$, the effective radius of the body. Show that model (b) assumes that $\displaystyle k_2$ is independent of the viscosity of air.

The bit in bold is the part I can't do - I got $\displaystyle k_1 \propto \mu r$ for the first bit. Thanks very much and any help is appreciated.

2. I haven't received a solution yet so I will explain my working and maybe someone will see a mistake. Square brackets denote the dimensions of the quantity, and M = mass, T = time, L = length:

$\displaystyle F = -k_2v^2$

$\displaystyle k_2 = -\frac{F}{v^2}$

So then $\displaystyle [k_2] = \frac{[F]}{[v]^2} = [M]\frac{[L]}{[T]^2}\cdot \frac{[T]^2}{[L]^2} = \frac{[M]}{[L]}$

Suppose $\displaystyle k_2 \propto \mu^{\alpha}r^{\beta}$

Then $\displaystyle [k_2] = [\mu]^{\alpha}[r]^{\beta}$

And $\displaystyle \frac{[M]}{[L]} = [M]^{\alpha}[L]^{- \alpha}[T]^{-\alpha}[L]^{\beta}$

$\displaystyle [M][L]^{-1} = [M]^{\alpha}[L]^{\beta - \alpha}[T]^{-\alpha}$

This gives us the equations $\displaystyle \alpha = 1, L = \beta - \alpha, - \alpha = 0$, which makes no sense, since how can $\displaystyle \alpha$ be zero and one? Help please !