Question

: A man walking briskly in rain with speed v must slant his umbrella forward making an angle θ with the vertical. A student
derives the following relation between θ and v : tan θ = v and checks that the relation has a correct limit: as v → 0, θ →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 ?

Answer 1

the answer is in the following image:-

Answer 2

Explanation:

The given relation:-

‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎tan Θ = V

Dimensionally ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎D [ L•H•S ] = D [ R•H•S ]

‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎dimensionless ≠ [ LT^(-1)]

Therefore , relation is not correct.

Correct relation must account for the speed of join drops also :-

‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎tan Θ = V/Vr , here RHS is also dimensionless.

‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎ ‎‎‎‎$\implies$ This formula appear to be Dimensionally correct.