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Escape Velocity

A variety of interesting misconceptions

The "Escape Velocity" of a planet is the speed with which an object would have to go to break free of the planet's gravity, assuming it was shot from a gun on the surface of the planet, and neglecting any other things such as atmosphere, rotation, etc.

The first problem with "escape velocity" is that many people assume that spacecraft have to go faster than the escape velocity to escape from the planet. This is not true. Spacecraft generally have engines which apply a force continuously, and this is in sharp contrast to the idea of blasting a cannonball out of a huge gun at a speed such that it will leave the planet in one go.

Another thing, though it may seem a picky detail, is that "escape velocity" is not a true velocity but a speed. Speed is how fast something is going, whereas velocity is speed in a direction.

To calculate the escape velocity of a planet, it's...

the square root of ((2 x G x M)/r)

...where
G is the universal gravitational constant 6.67 x 10^{11},
M is the mass of the planet in Kg, and r is the distance from the
centre of the planet (in metres). For the Earth, the escape
velocity is about 11,000 metres per second.

Because of the different gravity on different planets, it's easy to guess that the escape velocities are also different. However, it isn't always true that a higher force of gravity at the surface means it's going to have a higher escape velocity. The fact is that escape velocity has much more to do with the depth of the gravitational well than the force of gravity at the surface. For the initial assumptions of escape velocity to work, an object is fired at the escape velocity and its kinetic energy is enough to overcome the gravitational potential energy that it is owing by being inside the gravity well of the planet. Or, to put it another way...

½
x m x v^{2} = G x M x m / r

(Notice how m (the mass of the object) cancels out), leaving escape velocity as a function of the mass and size of the planet.

With this calculation applied to objects on the Earth, an energy of 62.5 megajoules per kilogramme is required to lift them out of the gravitational well of the planet.

Curiously, though, the energy required to go from the Earth into interstellar space is around four times this amount. The main gravity-well in question is that of the sun. So, even though the g-force due to the sun at earth's orbit is only about 3cm/s/s (much less than the 9.81m/s/s of the Earth), the gravity well of the sun is much bigger.

The energy required to go from floor "x" to floor "y" of a building could in theory be expressed as a speed in the same way as escape velocity, but it would seem silly. Although the "escape velocity" is not particularly relevant as a speed, it is relevant as an energy.

For minor planets and asteroids which have a very low gravitational pull, it may look initially as if it's possible to jump and escape, but for this to be true the planetesimal has to weigh less than 15 million million tonnes, a size equivalent to less than 9 kilometres in radius.

Another thing which is often bandied-about regarding escape velocity is the idea that if a collapsed star is so massive and compressed that its escape velocity exceeds the speed of light, then it's a black hole and "as nothing can go faster than light, nothing can escape from it". There's clearly a fault in this, as objects don't need to reach the escape velocity to leave. However, it's more likely that objects that are heavy enough to have become black holes have the spacetime around them so severely bent, and that's why things can't leave.