Question

The escape speed from Planet X is 20,000 m/s. Planet Y has the same radius as Planet X but is twice as dense. What is the escape speed from Planet Y?

Answer 1

Answer:

the escape speed from planet Y is $\sqrt{2}$ times the escape speed from planet X.

Explanation:

The escape speed from a surface of a planet is given by:

$v=\sqrt{\frac{GM}{R}}$

where

G is the gravitational constant

M is the mass of the planet

R is the radius of the planet

Let's call M the mass of planet X and R its radius. So the speed

$v_x=\sqrt{\frac{GM}{R}}$

corresponds to the escape speed from planet X.

Now we now that planet Y has:

- same radius of planet X: R' = R

- twice the density of planet X: d' = 2d

The mass of planet Y is given by

$M' = d' V'$

where V' is the volume of the planet. However, since the two planets have same radius, they also have same volume, so we can write

$M' = d' V= (2d)V = 2M$

which means that planet Y has twice the mass of planet X. So, the escape speed of planet Y is

$v'=\sqrt{\frac{GM'}{R}}=\sqrt{\frac{G(2M)}{R}}=\sqrt{2}(\sqrt{\frac{GM}{R}})=\sqrt{2} v$

so, the escape speed from planet Y is $\sqrt{2}$ times the escape speed from planet X.