In Gravity and Orbits you met the idea that an orbit is a fall that keeps missing the ground. With Newton's law of gravity we can find exactly how fast "fast enough" is, and how fast you must go to leave for good.
Why it works
Newton imagined a cannon on a very tall mountain. Fire a ball sideways and it falls to the ground. Fire faster and it lands farther away. At one special speed the ground curves away under the ball exactly as fast as the ball falls: it keeps falling and never lands. That is a circular orbit, where gravity provides the centripetal force: $\frac{GM}{r^2} = \frac{v^2}{r}$, so $v_c = \sqrt{\dfrac{GM}{r}}$. To escape, the kinetic energy must at least cancel the gravitational potential energy: $\frac{1}{2}mv^2 = \frac{GMm}{r}$, so $v_{\text{esc}} = \sqrt{\dfrac{2GM}{r}} = \sqrt{2}\,v_c$.
Key idea
Circular orbital speed $v_c = \sqrt{GM/r}$, with $r$ measured from the planet's center. Escape speed $v_{\text{esc}} = \sqrt{2GM/r} = \sqrt{2}\,v_c$. At Earth's surface: about $7.9$ km/s to orbit and $11.2$ km/s to escape.
Worked examples
Worked example: The space station
The International Space Station orbits about $400$ km up. Earth's radius is $6371$ km and $GM = 3.986 \times 10^{14}\,\text{m}^3/\text{s}^2$. Find its speed and period.
- $r = 6371 + 400 = 6771$ km $= 6.771 \times 10^{6}$ m.
- $v = \sqrt{\frac{3.986 \times 10^{14}}{6.771 \times 10^{6}}} = 7.67 \times 10^{3}$ m/s $= 7.67$ km/s.
- Period $= \frac{2\pi r}{v} = \frac{2\pi \times 6.771 \times 10^{6}}{7670} = 5547$ s, about $92$ minutes.
Answer: $7.67$ km/s, once around every $92$ minutes.
Worked example: Leaving the Moon
A spacecraft in a low circular orbit around the Moon moves at $1.68$ km/s. What speed would it need to escape from there?
- $v_{\text{esc}} = \sqrt{2}\,v_c = 1.414 \times 1.68$.
- $v_{\text{esc}} = 2.38$ km/s, far less than Earth's $11.2$ km/s.
Answer: $2.38$ km/s.
Common mistake
A spacecraft does not need to keep firing its engines to stay in orbit. Gravity at $400$ km is still about $90\%$ as strong as on the ground; the astronauts float because they and the station fall together. Moving sideways at $7.7$ km/s, the station needs no push to keep going (Newton's first law); it fires its thrusters only now and then to make up for the drag of the thin air. Check: the Moon has orbited Earth for billions of years with no engine at all.
Higher orbits are slower: geostationary satellites, $35{,}786$ km up, move at only $3.07$ km/s. Try launching from the mountain yourself in the Orbit Lab.