Q133 / 160 · AP EAMCET 2023 (15 May Shift 2) · Physics

The orbital velocity of a body near the surface of a planet ' $A$ ' is equal to escape velocity of a body from the planet ' $B$ '. If the masses of planets A and B are same, the ratio of their radii is

  • A
    $1$
  • Correct answer
    B
    $\frac{1}{2}$
  • C
    $\frac{1}{3}$
  • D
    $2$

Solution

Orbital velocity of a body of planet A, $\mathrm{V}_{\mathrm{A}}=\sqrt{\frac{\mathrm{GM}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{A}}}}$ Escape velocity of a body of planet B, $\begin{aligned} & \mathrm{V}_{\mathrm{B}}=\sqrt{\frac{2 \mathrm{GM}_{\mathrm{B}}}{\mathrm{r}_{\mathrm{B}}}} \\ & \mathrm{V}_{\mathrm{A}}=\mathrm{V}_{\mathrm{B}} \\ & \sqrt{\frac{\mathrm{GM}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{A}}}}=\sqrt{\frac{2 \mathrm{GM}_{\mathrm{B}}}{\mathrm{r}_{\mathrm{B}}}}\left(\because \mathrm{M}_{\mathrm{A}}=\mathrm{M}_{\mathrm{B}}\right) \\ & \frac{\mathrm{r}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{B}}}=\frac{1}{2} \end{aligned}$

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

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