Q. 254.5( 19 Votes )

# 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? If not, guess the correct relation.

Answer :

Dimensions of tan θ = M^{0}L^{0}T^{0}

(∵ All trigonometric functions have no units)

Dimensions for v (velocity) = M^{0}L^{1}T^{-1}

∴ The equation, tan θ = v is Dimensionally incorrect.

To balance the equation,

Let the speed of the rainfall be V, then we can make R.H.S dimensionless by dividing with V.

tan θ =

(∵ velocity dimension = M^{0}L^{1}T^{-1} always)

Now, the equation is dimensionally balanced and correct.

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