Question

In rain,a man Walk at a speed v and he must slant his umbrella at the angel theta with the vertical.the relation tan theta =v is derivid and check whether the limits c---o ,theta ---o are right . there is no strong wind and the raindrops fall vertically. Check whether the relation is correct, if not explain.

Answer 1

Answer:

Incorrect; on dimensional ground

The relation is tanθ=v

Dimension of R.H.S = M

0

L

1

T

−1

Dimension of L.H.S = M

0

L

0

T

0

( The trigonometric function is considered to be a dimensionless quantity)

Dimension of R.H.S is not equal to the dimension of L.H.S. Hence, the given relation is not correct dimensionally.

To make the given relation correct, the R.H.S should also be dimensionless. One way to achieve this is by dividing the R.H.S by the speed of rainfall v

′

Therefore, the relation reduces to

tanθ=v/v

′

This relation is dimensionally correct.

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.