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 $g=\frac{GM}{R^2},\frac{g_E}{g_M}=\frac{M_E}{M_M}\cdot \frac{R_M^2}{R_E^2}$
$\Rightarrow \frac{M_E}{M_M}=\left(\frac{g_E}{g_M}\right)\cdot {\left(\frac{R_E}{R_M}\right)}^2$
Now, $v=\sqrt{\frac{2GM}{R}}$
$\Rightarrow \frac{v_E}{v_M}={\sqrt{\frac{M_E}{M_M}\cdot \frac{R_M}{R_E}}}^{\prime }=\sqrt{\frac{6\times 10^2}{10}}=\sqrt{60}=7.75\approx 8.$