Question

The escape velocity 'v' of a body depends upon (1) the acceleration due to gravity (g) of the planet and (2) the radius (R) of the planet . Use the method of dimensions to obtain a relation between v,g and R.

Answer 1

this is your required answer

Answer 2

Answer:

$v=\sqrt{gR}$

Explanation:

Since  escape velocity 'v' of a body depends upon (1) the acceleration due to gravity (g) of the planet and (2) the radius (R) of the planet.

So,

$v =g^{a} R^{b}$.......................(1)

Since we know the dimension of velocity ,radius and acceleration due to gravity.

$v =LT^{-1}$,$R=L$ and $g=LT^{-2}$

Put the dimension of v,R and g in equation (1) we get,

$LT^{-1}=(L)^{a}(LT^{-2} )^{b}\\\\LT^{-1}=L^{a+b}T^{-2b}$

Compare the power of L and T we get,

a+b=1......................(2) and -2b=-1.........................(3)

From (3) we get

$b=\dfrac{1}{2}$

From (2) we get,

$a=\dfrac{1}{2}$

Put a and b in equation (1)

$v=\sqrt{gR}$