Answer to Question #103459 in Algebra for mm

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

$F=k\rho ^av^bA^c$

Here k is dimensionless constant, dimension of force $f$ is $MLT^{-2}$

dimension of density $\rho$ is $ML^{-3}$ , dimension of speed $v$ is $LT^{-1}$

and dimension of surface area $A$ is $L^2$ .

So, dimensionally we have: $MLT^{-2}=(ML^{-3})^a(LT^{-1})^b(L^2)^c$

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

$a=1,\;\;-3a+b+2c=1,\;\;-b=-2$

Solving this system we get: $a=1,\;b=2,\;c=1$

Therefore, the model will be: $F=k\rho v^2A$