Editing Potential energy (section)

== Potential energy for gravitational forces between two bodies ==

The gravitational potential function, also known as [[gravitational potential energy]], is:
<math display="block"> U=-\frac{GMm}{r}, </math>

The negative sign follows the convention that work is gained from a loss of potential energy.

=== Derivation ===

The gravitational force between two bodies of mass ''M'' and ''m'' separated by a distance ''r'' is given by [[Newton's law of universal gravitation]]
<math display="block">\mathbf{F}=-\frac{GMm}{r^2}\mathbf{\hat{r}},</math>
where <math>\mathbf{\hat{r}}</math> is [[Unit vector|a vector of length 1]] pointing from ''M'' to ''m'' and ''G'' is the [[gravitational constant]].

Let the mass ''m'' move at the velocity {{math|'''v'''}} then the work of gravity on this mass as it moves from position {{math|'''r'''(''t''<sub>1</sub>)}} to  {{math|'''r'''(''t''<sub>2</sub>)}} is given by
<math display="block"> W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r}\cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3} \mathbf{r}\cdot\mathbf{v} \, dt.</math>
The position and velocity of the mass ''m'' are given by
<math display="block">\mathbf{r} = r\mathbf{e}_r, \qquad\mathbf{v}=\dot{r}\mathbf{e}_\text{r} + r\dot{\theta}\mathbf{e}_\text{t},</math>
where '''e'''<sub>r</sub> and '''e'''<sub>t</sub> are the radial and tangential unit vectors directed relative to the vector from ''M'' to ''m''. Use this to simplify the formula for work of gravity to,
<math display="block"> W = -\int^{t_2}_{t_1} \frac{GmM}{r^3} (r\mathbf{e}_\text{r})\cdot(\dot{r}\mathbf{e}_\text{r} + r\dot{\theta}\mathbf{e}_\text{t})\,dt = -\int^{t_2}_{t_1}\frac{GmM}{r^3}r\dot{r}dt = \frac{GMm}{r(t_2)}-\frac{GMm}{r(t_1)}.</math>

This calculation uses the fact that
<math display="block"> \frac{d}{dt}r^{-1} = -r^{-2}\dot{r} = -\frac{\dot{r}}{r^2}.</math>