Two planets P₁ and P₂ with equal mass have radii R₁ and R₂, respectively, where R₂ = R₁/2. The escape speeds of P₁ and P₂ are v₁ and v₂, respectively. Then (v₂)/(v₁) is: (a) 2 (b) 1/√(2) (c) 1 (d) √(2)

Answer: (d). v ∝ √(M/R); equal masses so v ∝ 1/√(R) (v₂)/(v₁) = √((R₁)/(R₂)) = √(2) So the answer is (d) √(2).

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Escape Velocity

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

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Answer: (d)

$v \propto \sqrt{\frac{M}{R}}$ ; equal masses so $v \propto \frac{1}{\sqrt{R}}$
$\frac{v_2}{v_1} = \sqrt{\frac{R_1}{R_2}} = \sqrt{2}$
So the answer is (d) $\sqrt{2}$ .

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