A man walking briskly In the rain with speed v must slant his umbrella

1.	A man walking briskly In the 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 -> o, θ -& gt; o, 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.
Answer» Since tan θ is dimensionless, we need the RHS of the equation tan θ = v to be dimensionless. Let the speed of the rainfall be v¹. Then, by the method of dimensions, we get that tan θ = v/v¹ or v¹/v When the man is stationary (v = 0), he must hold the umbrella straight. i.e θ = 0° tan θ = 0 when v = 0 ∴ The required relation is tan θ = v/v¹

A man walking briskly In the 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 -> o, θ -& gt; o, 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.

Answer»

Since tan θ is dimensionless, we need the RHS of the equation tan θ = v to be dimensionless.

Let the speed of the rainfall be v¹. Then, by the method of dimensions, we get that tan θ = v/v¹ or v¹/v

When the man is stationary (v = 0), he must hold the umbrella straight.

i.e θ = 0°

tan θ = 0 when v = 0

∴ The required relation is tan θ = v/v¹