Question

Suppose you are driving a van down a highway. Use dimensional analysis to find the wind force
you are experiencing, assuming that the force is affected by the wind density, the speed of the van
and its surface area exposed to the wind direction.

Answer 1

Assumptions: the force F of the wind is affected by the speed v of the van and the surface area A of the van directly exposed to the wind’s direction.

Hypothesis: the force is proportional to some power of the speed times some power of the surface area. The strength of the wind is also affected by its density (density of air): 

$$F=kv^aA^b\rho^c$$

Here $k$ is dimensionless constant, dimension of force $F$ is $MLT^{-2},$ dimension of speed $v$ is $LT^{-1},$ dimension of surface area $A$ is $L^2,$ and dimension of density $\rho$ is $ML^{-3},$

So, dimensionally we have: 

$$MLT^{-2}=(M^0L^0T^0)(LT^{-1})^a(L^2)^b(ML^{-3})^c$$

Equating the exponents on both sides of this equation leads to the system of linear equations:

$$1=c$$ $$1=a+2b-3c$$ $$-2=-a$$

Solving this system we get: 

$$a=2$$ $$b=1$$ $$c=1$$

Therefore, the model will be:

$$F=kv^2A\rho$$

Note: for a particular situation, the density $\rho$ is a constant (but this constant has dimension).