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
B. `v^(2) prop g lambda`
C. `v^(2) prop g lambda^(2)`
D. `v^(2) prop g^(-1)lambda^(2)`

Answer

B

Explanation

`v=k lambda^(a) rho^(b)g^(c)`
`[M^(0)LT^(-1)]L^(a)(ML^(-3))^(b){LT^(-2))^(c)=M^(b)L^(a-3b+c)T^(-2c)`
`therefore" "b=0, a-3b+c=1`
`-2c=-1 rArr c=1//2" "therefore" "a=(1)/(2)`
`v prop lambda^(1//2) rho^(0) g^(1//2) or v^(2) prop lambdag`