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.

The escape velocity v of a body depends upon (i) the acceleration due

1.	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.
Answer» Solution :`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)`