Gravitational potential energy is energy an object possesses because of its position in a gravitational field. The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass \(\:m \:\) to a given point in space. The gravitational force between the earth of mass \(\:M_e \:\) and a mass \(\:m \:\) separated by a distance \(\:r \:\) is given by
$$
F=\frac{GM_em}{r^2}
$$
The gravitational potential energy, \(\: U\:\) is given by
$$
U=-\int_{\infty}^{r}Fdr = -\int_{\infty}^{r}\frac{GM_em}{r^2}dr
$$
$$
= -GmM_e\int_{\infty}^{r}r^{-2}dr = -GmM_e\left(-\frac{1}{\infty} – \frac{1}{r} \right)
$$
$$
U = \frac{GmM_e}{r}
$$
The gravitational Potential (\(\: V\:\)) is defined as the gravitational potential energy of unit mass \(\: m\:\). Hence,
$$
V=\frac{U}{m} =\frac{\frac{GmM_e}{r}}{m}$$
$$V=-\frac{GM_e}{r}$$
Gravitational potential is a scalar quantity and its SI unit is J/kg.
