Escape Velocity

The gain of the potential energy of an object inside the gravitational field of another object between a radius $r$ and a radius $R$ can be calculated according to $$E_\mathrm{pot} = \int_r^R G\frac{Mm}{r^2}\,\mathrm{d}r$$ The full potential energy at a given radius $R$ is then given as the integral from $R$ to infinity: $$E_\mathrm{pot} = \int_r^\infty G\frac{Mm}{r^2}\,\mathrm{d}r$$ This is the work, which would be necessary, to bring an object from the given point $r$ to a place that is infinitely far away. Solving this integral leads to: $$\boxed{E_\mathrm{pot} = G\frac{Mm}{r}}$$ Using the law of energy conservation can be used to calculate the initial speed which is necessary for that: $$\frac{1}{2}mv^2 = G\frac{Mm}{r}$$ The solution of that equation is called escape velocity and can be written as $$\boxed{v = \sqrt{\frac{2GM}{r}}}$$ Is is by the factor $\sqrt{2}$ larger than the orbtial velocity.