Q110 / 180 · NEET 2026 (Re-NEET) · Physics

Two planets $P_1$ and $P_2$ with equal mass have radii $R_1$ and $R_2$, respectively, where $R_2=\dfrac{R_1}{2}$. The escape speeds of $P_1$ and $P_2$ are $v_1$ and $v_2$, respectively. Then $\dfrac{v_2}{v_1}$ is :

  • A
    $2$
  • B
    $\dfrac{1}{\sqrt{2}}$
  • C
    $1$
  • Correct answer
    D
    $\sqrt{2}$

Solution

The escape speed of a planet of mass $M$ and radius $R$ is given by $v = \sqrt{\dfrac{2GM}{R}}$.

For planet $P_1$, the escape speed is $v_1 = \sqrt{\dfrac{2GM}{R_1}}$.

For planet $P_2$, the escape speed is $v_2 = \sqrt{\dfrac{2GM}{R_2}}$.

Taking the ratio of the escape speeds:

$\dfrac{v_2}{v_1} = \sqrt{\dfrac{R_1}{R_2}}$

Given that $R_2 = \dfrac{R_1}{2}$, substituting this into the equation gives:

$\dfrac{v_2}{v_1} = \sqrt{\dfrac{R_1}{\dfrac{R_1}{2}}} = \sqrt{2}$

Answer: $\sqrt{2}$

Hence, the correct answer is option: 'D'.

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