JEE Main 2023 — Gravitation Question with Solution

The formula to calculate the escape velocity is given by

$v_e=\sqrt{\frac{2GM}{R}} ...\left(1\right)$

For the Earth, it is given by

${v_e}^E=\sqrt{\frac{2G{M}_E}{R_E}}...\left(2\right)$

 and, for the planet, it is given by

${v_e}^P=\sqrt{\frac{2G{M}_P}{R_P}}...\left(3\right)$

Divide equation (2) by equation (3) and simplify to obtain the required ratio.

$\frac{{v_e}^E}{{v_e}^P}=\sqrt{\frac{M_E}{M_P}}\times \sqrt{\frac{R_P}{R_E}} ...\left(4\right)$

Substitute the values of the known parameters into equation (4) to obtain the required ratio.

$\begin{array}{rcl}\frac{{v_e}^E}{{v_e}^P}& =& \sqrt{\frac{M_E}{16M_E}}\times \sqrt{\frac{4R_E}{R_E}}\\ & =& \frac{1}{2}\\ & \Rightarrow & {v_e}^P:{v_e}^E= 2:1\end{array}$