The velocity of sound `(upsilon)` in a gas depends upon coefficint of volume elesticity E of the gas and density d of the gas. Use method of dimensions to derive the formula for `upsilon.`

Explanation

Let V=`kE^a d^b ` where k is proportionality constant.
Dimesnional formula of v, E and d are
`V to [M^0 LT^(-1)] , E to [ML^(-1)T^(-2)] , d to [ML^(-3)T^0]`
As k has no dimensions
`[M^0 LT^(-1)] = [ML^(-1)T^(-2)]^a [ML^(-3)T^0]^b`
`therefore [M^0 LT^(-1)] = [M]^(a+b) [L]^(-a-3b) [T]^(-2a)`
equation dimensions of M, L and T on either side of the equation.
We get , a+b =0 , -a-3b =1
`-2a=-1`
`therefore a=1//2 , a+b =0`
`therefore b=-1//2 `
`therefore V =KE^(1//2)d^(-1//2) , V=ksqrt(E/d)`
experimentally the value of k is found to be one.
`therefore V= sqrt(E/d)`