Newton drew a picture of it. Set a cannon on a high mountain and fire sideways. A weak shot lands near, a stronger one lands farther off.
Stronger still and the landing point swings round past the horizon. Beyond a certain speed it never reaches the ground at all, because it keeps falling and the ground keeps curving away by as much. That is a circular orbit.
Stronger again and the path becomes an ellipse, going far out and still returning. And beyond one further speed it never returns.
There is no break between falling and orbiting. Raising the speed alone carries you continuously from one to the other.
Write down the condition for never returning: arriving infinitely far away with exactly no speed left. That is the dividing line.
There, both the energy of motion and the energy of position are . The total does not change, so it must have been at the start as well: .
Solving gives , the escape velocity.
Sending the semi-major axis to infinity in the vis-viva equation of the previous article gives the same thing. Against the speed that holds a circle, it is larger by exactly .
Only the distance appears in the formula. Which way you threw it does not.
Straight up, sideways, or at any angle between, the speed required is the same, because energy depends on the square of the speed and carries no direction.
The paths are entirely different: a straight line upward, a strongly curved hyperbola sideways. The dividing line between returning and not returning is the same number either way.
This is all for a single push followed by nothing. Something that can keep pushing gets away however slowly it goes. An elevator to space would not need km per second.
At the surface of the Earth, km per second. At the Moon, . The gravity is a sixth and the radius smaller, so it comes out far below the Earth.
At the surface of the Sun, km per second. Squeezed to a white dwarf, ; to a neutron star, , approaching half the speed of light.
Squeeze further, until the escape velocity reaches the speed of light, and that surface is the event horizon. Not even light gets out, and beyond it there is no route to the outside. The phrase black hole comes out of that single line.
Michell made this estimate in , calculating naively from the inverse square. General relativity returned the same radius years later.
The escape velocity also decides whether a body can hold on to an atmosphere.
Gas molecules do not all move at one speed. Some are fast and some are slow, spread over a range. Where the fast tail of that range reaches past the escape velocity, those molecules leave.
On the Earth the tail does not reach. So there is air. On the Moon it does. So there is none.
Hydrogen and helium are light and therefore fast at the same temperature. The Earth cannot hold either, so the two commonest elements in the universe are almost absent from its atmosphere.