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 $R$ .

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Solution

Step 1: Given Data
The escape velocity $v$ depends on Acceleration due to gravity $(g)$ and The radius $(R)$ of the planet
Step 2: Dimensional analysis
If the relation is $v = g^{a} R^{b}$
The dimensions of the LHS and RHS have to be the same.
By plugging in the dimensional formulas of the respective variables in the equation we can get the values of $a$ and $b$
Step 3: Derive the Dimensions
DImensional formula of $v$ is $M^{0} L^{1} T^{- 1}$
Dimensional formula of $g$ is $M^{0} L^{1} T^{- 2}$
Dimensional formula of $R$ is $M^{0} L^{1} T^{0}$
⇒ $M^{0} L^{1} T^{- 1} = {[M^{0} L^{1} T^{- 2}]}^{a} {[M^{0} L^{1} T^{0}]}^{b}$
⇒ $M^{0} L^{1} T^{- 1} = M^{0} L^{a + b} T^{- 2 a}$
Step 4: Solving for $a a n d b$
⇒ $a + b = 1 & - 2 a = - 1$
⇒ $a = \frac{1}{2}$
⇒ $b = \frac{1}{2}$
Thus, the value of both $a$ and $b$ is $\frac{1}{2}$ and $v = C \sqrt{g R}$ where $C$ is some constant that can be calculated experimentally.

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