The escape velocity from the surface of the Earth is \(11.2 \mathrm{~km} / \mathrm{s}\). Find the escape velocity from a…

Escape velocity (\({v}_{e}\)​) is given by:

\({v}_{e}=\sqrt{\frac{2GM}{R}}\)

where:

• \(G\) is the gravitational constant.
• \(M\) is the mass of the planet.
• \(R\) is the radius of the planet.

For Earth:

\({v}_{e,\text{Earth}}=11.2\text{ km/s}\)

The escape velocity of a planet compared to Earth is:

\(\frac{{v}_{e,\text{Planet}}}{{v}_{e,\text{Earth}}}=\sqrt{\frac{{M}_{\text{Planet}}\mathrm{/}{M}_{\text{Earth}}}{{R}_{\text{Planet}}\mathrm{/}{R}_{\text{Earth}}}}\)

• The mass of the planet is 8 times the mass of Earth: \({M}_{\text{Planet}}=8{M}_{\text{Earth}}\)​
• The radius of the planet is 2 times the radius of Earth: \({R}_{\text{Planet}}=2{R}_{\text{Earth}}\)​

\(\frac{{v}_{e,\text{Planet}}}{{v}_{e,\text{Earth}}}=\sqrt{\frac{8{M}_{\text{Earth}}}{2{R}_{\text{Earth}}}}\)​ \(=\sqrt{\frac{8}{2}}=\sqrt{4}=2\)

​ \({v}_{e,\text{Planet}}=2\times 11.2=22.4\text{ km/s}\)

\(22.4\text{ km/s}\)​