$A$ man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle $\theta$ with the vertical. $A$ student derives the following relation between $\theta$ and $v: \tan \theta = v$ and checks that the relation has a correct limit: as $v \rightarrow 0, \theta \rightarrow 0,$ as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not,guess the correct relation.

(A) The relation $\tan \theta = v$ is incorrect.
Dimensional Analysis:
Dimension of $L.H.S = \tan \theta = [M^0 L^0 T^0]$ (dimensionless).
Dimension of $R.H.S = v = [L T^{-1}]$.
Since the dimensions of $L.H.S$ and $R.H.S$ do not match,the equation is dimensionally inconsistent.
To make it dimensionally correct,the $R.H.S$ must also be dimensionless. This can be achieved by dividing $v$ by the speed of the rain $v'$,which is also a velocity.
Therefore,the correct relation is $\tan \theta = \frac{v}{v'}$,where $v'$ is the speed of the rain.