__Escape Velocity__

Escape velocity is the velocity that something must be going to be able to
pull free of the gravitational field of another body. For example, if you
want to go to the moon, you need to achieve escape velocity in order to do it.

We can calculate escape velocity by starting with Newton's law of
gravity:

- F

Here F

G is the gravitational constant = 6.6726

- dE = GM

dx is the tiny distance moved (so that the force of gravity is essentially a constant) and dE is the tiny amount of energy it took to move the bodies apart by dx.

If we could add all the tiny dxs up as the bodies move apart (to essentially
an **infinite** distance apart) we would know the energy we'd have to impart
to the bodies (this is known as their **Gravitational Potential Energy**) in
the first place in the form of kinetic energy to allow them to achieve escape
velocity.

How do you add an **infinite** number of dxs?
What seems impossible here is accomplished by a mathematical wonder called
integration. Isaac Newton was the first to do this (Gottfried Leibnitz did it
a bit later, independently of Newton) and it is one of the reasons for his
great fame. For a more detailed discussion of how integration works,
click here

When the above equation is integrated, we get:

- E = GM

evaluated between r and infinity.

Now that we have the energy required we can calculate the escape velocity.
The energy of a moving body is given by its kinetic energy:

- E

E

To achieve escape velocity, the kinetic energy must equal the gravitational
potential energy, so we set our equations equal to each other and solve for
V. We'll also assume that M_{1} is very small, and M_{2} is
very large. By doing this, we can ignore the effect that M_{1}
has on M_{2}, which simplifies the math considerably.

- M

The M

V

Just for fun, how fast does a rocket have to go to escape earth's
gravitational field? When we plug in the numbers, we get:

- V = sqrt(2 x 6.6726 x 10

(That's about 25,000mph, for the metrically impaired)

__The Schwartzchild Radius
__

We just rearrange the last equation so we are solving for r, not V:

- r = 2GM

- r = 2GM

Note that our derivation did not mention the word relativity once. Yet using the full blown theory of General Relativity, with its attendant tensor complexity, Karl Schwartzchild was able to derive the above. The fact that classical methods and the theory of General Relativity arrive at the same conclusion is, to say the least, remarkable.