GATE 2023 AE – Question 62
A planet has mass $6.4169\times10^{23}$ kg and radius 3390 km. For $G=6.67\times10^{-11}\ \mathrm{Nm^2/kg^2}$, escape speed (km/s, one decimal place) is _____.
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Correct answer: 4.97 to 5.03
Explanation
**Escape speed** from the surface of a planet of mass $M$ and radius $R$ (the speed at which the kinetic energy equals the gravitational potential energy):
$$\tfrac12V^2=\frac{GM}{R}\;\Rightarrow\;V_{esc}=\sqrt{\frac{2GM}{R}} .$$
**Numbers.** $GM=6.67\times10^{-11}\times6.4169\times10^{23}=4.2802\times10^{13}\ \text{m}^3/\text{s}^2$ and $R=3.39\times10^6$ m:
$$V_{esc}=\sqrt{\frac{2\times4.2802\times10^{13}}{3.39\times10^6}}=\sqrt{2.5252\times10^7}=5025\ \text{m/s}.$$
The escape speed is **5.0 km/s**.