The velocity of water wave `v` may depend on their wavelength `lambda`, the density of water `rho` and the acceleration due to gravity `g`. The method of dimensions gives the relation between these quantities as

Options

A. `V^2 prop g lambda`
B. `V^2 prop 1/(g lambda)`
C. `V^2 prop 1/(gd)`
D. `V^2 lambda g lambda d`

Answer

A

Explanation

`V prop lambda^a d^b g^c`
`[V] = [lambda]^a [d]^b [g]^c`
`[M^0LT^(-1)] = [L]^a [ML^(-3)]^b [LT]^c`
`b = 0, " " a - 3b + c = 1 , " " - 2c = -1`
`c = 1/2, a = 1 -1/2 = 1/2`
`V prop lambda^(1//2) g^(1//2) implies V^(2) prop lambda g`