Question

if the Earth is 81 times heavier and 4 times larger in size than the moon then show that the escape velocity on the moon is 2/9 times that on the Earth​

Answer 1

Earth is 81 times heavier and 4 times larger in size than the moon. Let escape velocity of earth be v_(e).

$\sf v_{e} = \sqrt{\dfrac{2GM}{R}}$

Now, mass of moon = M/81 and radius of moon = R/4.

$\sf v_{m} = \sqrt{\dfrac{2G \frac{M}{81}}{ \frac{R}{4}}}$

$\sf \implies v_{m} = \sqrt{ \dfrac{4}{81} \times \dfrac{2G M}{R}}$

$\sf \implies v_{m} = \dfrac{2}{9} \times \sqrt{ \dfrac{2G M}{R}}$

$\sf \implies v_{m} = \dfrac{2}{9} \times v_{e}$

[Hence proved]