Circular Orbits and Escape Speed

Newton's cannon, circular velocity, free fall and escape speed

De mundi systemate, 1687

Newton answered the question himself, with a drawing. Put a cannon on a mountain so tall it pokes above the atmosphere. Fire it horizontally. The ball follows a curve and lands some distance away. Add powder: it lands further. Add more: further still. Now keep going. Because the Earth is round, the ground is curving away beneath the ball the entire time it falls. Add enough powder and you reach a speed where the ball falls toward the Earth at precisely the rate the Earth's surface curves away from underneath it. The ball never gets any closer to the ground. It never stops falling either. It just keeps falling, forever, around and around. That is an orbit. An orbit is not a way of escaping gravity. It is a way of falling and continuously missing.

Watch first · presented by Shoaib Aziz

Every trajectory in the cannon sequence is a real conic computed from its launch speed, and every number on screen is one the labs compute for themselves — 7.67 km/s, the 88% gravity at the space station and the √2 escape ratio.

There is an art, it says, or rather, a knack to flying. The knack lies in learning how to throw yourself at the ground and miss.

– Douglas Adams, Life, the Universe and Everything, 1982

Figure 1.3

Newton's cannon, computed rather than sketched: every curve here is a real conic section traced from the actual equations of motion. Below circular speed the ball strikes the ground. At exactly circular speed it comes back to where it started. Above it, the orbit stretches into an ellipse — until at √2 times circular speed it never returns at all.

Let's put a number on 'enough powder'. Earth's radius is 6371 km, and a little geometry says that over a horizontal distance of about 8 km, the surface drops away by roughly 4.9 metres.

Now — how far does anything fall in one second under gravity?

\tfrac{1}{2}gt^2 = \tfrac{1}{2}(9.81)(1)^2 = 4.9

metres.

The same number. So if you travel 8 km horizontally in one second, you fall 4.9 m and the ground drops 4.9 m, and your altitude is unchanged.

Figure 1.4

Travel about 8 km horizontally and Earth's surface falls away by 4.9 m — exactly the distance you fall in one second. Match those two rates and you are in orbit. (Vertical scale hugely exaggerated.)

That was the back-of-an-envelope version. Now the real derivation, which takes two lines.

For a circular orbit, the acceleration required to keep turning is the centripetal acceleration

v^2/r

. The acceleration available is gravity,

\mu/r^2

. A circular orbit is precisely the case where the available exactly equals the required:

\frac{v^2}{r} = \frac{\mu}{r^2} \;\;\Longrightarrow\;\; v_{c} = \sqrt{\frac{\mu}{r}}

Circular orbital speed. One r cancels, and the single most-used formula in astrodynamics falls out.

💡Higher orbits are slower

Look at the formula again:

v_c = \sqrt{\mu/r}

. Speed goes down as radius goes up. This catches almost everyone. The ISS at 408 km altitude screams along at 7.67 km/s. GPS satellites at 20,200 km amble at 3.87 km/s. Geostationary satellites at 35,786 km barely manage 3.07 km/s. And this has a consequence that will haunt Unit 6: to catch up with something ahead of you in the same orbit, you have to slow down. Speeding up raises your orbit, which makes you slower, and you fall behind. Chasing in space works backwards from every intuition you have.

📐Worked Example: the ISS

Altitude 408 km. Radius from Earth's centre:

r = 6371 + 408 = 6779

v_c = \sqrt{398600 / 6779} = \sqrt{58.80} = \mathbf{7.67\ km/s}

That's about 27,600 km/h — faster than a rifle bullet, and enough to cross a country in a couple of minutes. One full orbit takes 92 minutes. Note what we did with the altitude: we added it to Earth's radius. Orbital radius is always measured from the centre of the body, never from the surface. This is the single most common arithmetic mistake in this subject.

Now push the speed past circular. The ball no longer comes back to the same altitude — it climbs, slows, and swings out into an ellipse before returning. Push harder and the ellipse gets longer. Keep pushing and you reach a threshold where it never comes back at all:

v_{esc} = \sqrt{\frac{2\mu}{r}} = \sqrt{2}\;v_{c}

Escape speed — always exactly √2 ≈ 1.414 times the local circular speed, at every altitude, around every body.

💡Escape speed is √2 times circular speed

That

\sqrt{2}

is a remarkably cheerful fact. Once you are in a circular orbit, you need only 41% more speed to leave the planet entirely. Getting to orbit in the first place is the brutal part — that's roughly 9.4 km/s of effort once you include atmospheric drag and gravity losses. Leaving from there costs another 3.2 km/s. As the saying goes: low Earth orbit is halfway to anywhere.

One last thing to clear up, because it is the most persistent misconception in all of spaceflight. Astronauts on the ISS are not in zero gravity. We computed it in the last lesson: at 408 km,

g = \mu/r^2 = 8.66

m/s², which is 88% of what you feel standing on the ground.

They float because they are in free fall — and so is the station around them, falling at exactly the same rate. There is no floor pressing up on them. What we call 'weight' was never gravity; it was the floor.

Figure 1.5

Gravity is essentially identical in all three situations. The only thing that changes is whether something is pushing back. "Weightlessness" is the absence of a normal force, not the absence of gravity.

Orbit	Altitude (km)	Radius r (km)	Speed (km/s)	Period
ISS (LEO)	408	6779	7.67	92 min
Starlink	550	6921	7.59	95 min
GPS (MEO)	20,200	26,571	3.87	11 h 58 m
GEO	35,786	42,157	3.07	23 h 56 m
Moon	384,400	384,400	1.02	27.3 days

Every speed here is just √(μ/r). One formula covers the entire table — from a space station to the Moon.

↓But there is a problem

So we've explained the circle. There is only one problem with that achievement: nothing in the solar system actually moves in a circle. Not one planet, not one moon, not one comet, not one satellite. Kepler knew this a generation before Newton was born. Proving it cost him a decade of his life and nearly his sanity — and the answer he found was so unwelcome that he threw it away twice before accepting it.