Question

The escape speed from Planet X is 10,000 m/s. Planet Y has the same radius as Planet X but is twice as dense. What is the escape speed from Planet Y?

Short answer

Escape velocity of Planet Y is 14140 m/sec

Step-by-step solution

Given Information

Escape velocity of planet X, is 10000 m/s
Density of planet Y is twice the density of planet X => ρ_x = 2ρ_y

Radius of planet Y is equal to the radius of planet X => R_x = R_y

Explanation

We can calculate escape velocity as

$v_e=\sqrt{\frac{2GM}{r}}$

Where G = universal gravitational constant, M = mass of the planet
r = radius of the planet

The escape velocity of planet X is

$v_X=\sqrt{\frac{2GM}{R_X}}=\sqrt{\frac{2G{\rho }_{X}V}{R_X}}(\text{as}M=\rho \times V)$

And for plant Y is

$v_Y=\sqrt{\frac{2GM}{R_Y}}=\sqrt{\frac{2G{\rho }_{Y}V}{R_Y}}(\text{as}M=\rho \times V)$

Now get the ratio and substitute values given

$$\begin{gathered} \frac{v_X}{v_Y}=\frac{\sqrt{\frac{2G{\rho }_{X}V}{R_X}}}{\sqrt{\frac{2G{\rho }_{Y}V}{R_Y}}} \\ \frac{10000m/s}{v_Y}=\sqrt{\frac{{\rho }_X}{2{\rho }_X}} \\ {v}_Y=14140\mathrm{m}/\mathrm{s} \end{gathered}$$