Question

The ratio of the radius of the Earth to that of moon is $10. $ The ratio of acceleration due to gravity on Earth and on Moon is $6$ . The ratio of the escape velocity from the surface of Earth to that from the Moon is

• A. 10
• B. 5
• C. nearly 8
• D. 1.66

Answer 1

Answer: (C)

$\frac{R_{E}}{R_{M}}=10, \frac{g_{E}}{g_{M}}=6$
But $g=\frac{G M}{R^{2}}, \frac{g_{E}}{g_{M}}=\frac{M_{E}}{M_{M}} \cdot \frac{R_{M}^{2}}{R_{E}^{2}}$
$\Rightarrow \frac{M_{E}}{M_{M}}=\left(\frac{g_{E}}{g_{M}}\right) \cdot\left(\frac{R_{E}}{R_{M}}\right)^{2}$
Now, $v=\sqrt{\frac{2 G M}{R}}$
$\Rightarrow \frac{v_{E}}{v_{M}}=\sqrt{\frac{M_{E}}{M_{M}} \cdot \frac{R_{M}}{R_{E}}}^{\prime}=\sqrt{\frac{6 \times 10^{2}}{10}}=\sqrt{60}=7.75 \approx 8 .$