Answer:
the escape speed from planet Y is [tex]\sqrt{2}[/tex] times the escape speed from planet X.
Explanation:
The escape speed from a surface of a planet is given by:
[tex]v=\sqrt{\frac{GM}{R}}[/tex]
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
[tex]v_x=\sqrt{\frac{GM}{R}}[/tex]
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
[tex]M' = d' V'[/tex]
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
[tex]M' = d' V= (2d)V = 2M[/tex]
which means that planet Y has twice the mass of planet X. So, the escape speed of planet Y is
[tex]v'=\sqrt{\frac{GM'}{R}}=\sqrt{\frac{G(2M)}{R}}=\sqrt{2}(\sqrt{\frac{GM}{R}})=\sqrt{2} v[/tex]
so, the escape speed from planet Y is [tex]\sqrt{2}[/tex] times the escape speed from planet X.
