Question 13.8: Escape Speed of a Rock Superman picks up a 20-kg rock and hu...
Escape Speed of a Rock
Superman picks up a 20-kg rock and hurls it into space. What minimum speed must it have at the Earth’s surface to move infinitely far away from the Earth?
Conceptualize Imagine Superman throwing the rock from the Earth’s surface so that it moves farther and farther away, traveling more and more slowly, with its speed approaching zero. Its speed will never reach zero, however, so the rock will never turn around and come back.
Categorize This example is a substitution problem.
Use Equation 13.22 to find the escape speed:
\begin{aligned}v_{\text {esc }} & =\sqrt{\frac{2 G M_E}{R_E}}=\sqrt{\frac{2\left(6.674 \times 10^{-11} N \cdot m^2 /kg^2\right)\left(5.97 \times 10^{24} kg\right)}{6.37 \times 10^6 m}} \\& =1.12 \times 10^4 m/s\end{aligned}The calculated escape speed corresponds to about 25 000 mi/h. The mass of the rock does not appear in the calculation. Therefore, this is also the escape speed for Superman throwing a 5 000-kg spacecraft from the surface of the Earth. Furthermore, if a spacecraft is in an orbit around the Earth, its orbital radius r is close to that of the Earth, R_E, so the escape speed we have calculated is also valid for the non-superhero situation of a spacecraft in orbit firing its engines to escape that orbit.