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^(b)rArr v = k g^(a)R^(b), K to` dimensionless proportionality 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`
`:. "v k" sqrt(gR)`