Q 2.25: 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.

Answers (1)

The derived formula tan\Theta = v is dimensionally incorrect.

We know, Trigonometric functions are dimensionless. 

Hence , [tan\Theta ] = M^0L^0T^0

and [v] = M^0L^1T^{-1} = LT^{-1}.

\therefore To make it dimensionally correct, we can divide v by v_{r} (where v_{r} is the speed of rain)

Thus, L.H.S and R.H.S are both dimensionless and hence dimensionally satisfied.

The new formula is : tan\Theta = v / v_{r}

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