The escape vleocity v of a body depends upo (i) the acceleration due to gravity of the planet and (ii) the radius of the planet R. Establish dimesionally the releaitonship betweeeen v, g and R.

Explanation

`valphag^(a)R^(b)impliesv=kg^(a)R^(b),Krarr` 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=-2aimpliesa=(1)/(2)`
`b=1-a=1-(1)/(2)=(1)/(2)`
`therefore v=ksqrt(gR)`