The escape velocity v of a body depends upon (i) the acceleration due to gravity of the planet and (ii) the radius of the planet R. Establish dimensionally the relationship between v, g and R.

Explanation

`v prop g^a R^h rArr v = k g^a R^h , K to` dimensionless proportinally constant
`[v] = [g]^a[R]^b`
`[M^0 L^1 T^(-1) ] = [M^0 L^1 T^(-2) ]^a [ M^0 L^1 T^0]^b`
equating powers
`1 = a+ b`
` - 1 = -2a rArr a = 1/2`
` b = 1 - a = 1 - 1/2 = 1/2`
` therefore v = k sqrt(gr)`