Dimensions

**slevvio** · September 29th 2008, 05:36 AM

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

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

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

**slevvio** · October 3rd 2008, 09:24 AM

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:

$F = -k_2v^2$

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

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

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

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

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

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

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

!

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