## Getting Out of Earth

The work required to lift an object away from the surface of the Earth is given by the force of gravity integrated over the distance. (From the surface of Earth to a point in space where the object will not fall back to Earth.)
Mathematically written as :

W = (Sum from Rf to RE) F dr

= (Sum from Rf to RE) G (m ME / r2) dr

= G m ME (Sum from Rf to RE) dr / r2

 Rf = point in space RE = point on the surface of Earth ME = mass of Earth r = distance from Rf to RE G = Gravitational constant Uf = Potential energy at Rf UE = Potential energy at RE K = Kinetic energy needed to go from RE to Rf

The work is the change in potential energy :

W = [-G m ME / Rf] - [-G m ME / RE]

W = Uf - UE

If the vessel is lifted to infinity, it has a potential energy of zero.

W = Uf - UE

= 0 - UE

= -UE

It means that back on the surface of the Earth, it's potential energy is negative.

W = -UE = -G m ME / RE

To escape from the Earth it must be given some kinetic energy enough so that it won't slow down to zero before it gets infinitely far away.

K = 1/2 mv2

The total energy is zero both at infinity and on the surface of Earth.

-G m ME / RE + 1/2 mv2 = 0

Solving this for v (velocity) :

1/2 mv2 = G m ME / RE

v2 = 2 G ME / RE

v = (2 G ME / RE)^(1/2)

v = 11 km / s

This velocity is called the escape velocity . A vessel will escape if it starts with the speed of about 11 km per second .