Question 303: Gravitation

Question

The radius of the planet is double that of the earth, but their average densities are same. $\mathrm{V}_{\mathrm{p}}$ and $V_E$ are the escape velocities of planet and earth respectively. If $\frac{V_P}{V_E}=x$, the value of ' $x$ ' is

Accepted Answer

Escape velocity is given by,
$$\begin{aligned}
\mathrm{v}_{\mathrm{c}} & =\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}} \
& =\sqrt{\frac{2 \mathrm{G}}{\mathrm{R}} 	imes \frac{4}{3} \pi \mathrm{R}^3 ho}=\sqrt{\frac{8 \mathrm{G}}{3} \pi \mathrm{R}^2 ho}=\sqrt{\frac{8 \mathrm{G} \pi ho}{3}} 	imes \mathrm{R}
\end{aligned}$$
As the planets have the same density,
$$\begin{aligned}
& \mathrm{V}_{\mathrm{e}} \propto \mathrm{R} \
& \frac{\mathrm{~V}_{\mathrm{P}}}{\mathrm{~V}_{\mathrm{E}}}=\frac{\mathrm{R}_{\mathrm{P}}}{\mathrm{R}_{\mathrm{E}}}=\frac{2 \mathrm{R}}{\mathrm{R}}=2 \
	herefore \quad & \frac{\mathrm{~V}_{\mathrm{P}}}{\mathrm{~V}_{\mathrm{E}}}=2 \
	herefore \quad & X=2
\end{aligned}$$

Answer

$\frac{1}{4}$

Answer

$\frac{1}{2}$

Answer

2

Answer

4

Gravitation

The radius of the planet is double that of the earth, but their average densities are same. \mathrm{V}_{\mathrm{p}} and V_E are the escape velocities of planet and earth respectively. If \frac{V_P}{V_E}=x, the value of ' x ' is

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