Gravitational potential energy (PE) is the energy a body possesses because of its position in a gravitational field. Near the Earth's surface we use the familiar $U=mgh$, but that is only an approximation valid for small heights where $g$ is nearly constant. For larger distances we must account for the fact that gravity weakens with distance.
Taking the reference point at infinity (where the gravitational force is zero), the work done by gravity in bringing a mass $m$ from infinity to a distance $r$ from a mass $M$ gives the potential energy:
- $U=-\frac{GMm}{r}$.
- The negative sign shows the field is attractive — the system has less energy than when the masses were infinitely far apart. PE is maximum (zero) at infinity and becomes more negative as the bodies come closer.
Gravitational potential ($V$) is the potential energy per unit mass at a point: it is the work done in bringing a unit mass from infinity to that point.
- $V=-\frac{GM}{r}$, with SI unit $\text{J/kg}$.
- Relationship: $U=mV$. Like PE, the potential is negative and tends to zero at infinity.
Escape velocity ($v_e$) is the minimum speed with which a body must be projected from the surface of a planet so that it just escapes the planet's gravitational pull and never returns. To escape, the body's kinetic energy must at least equal the magnitude of its gravitational potential energy at the surface. Setting $\frac{1}{2}mv_e^2=\frac{GMm}{R}$ gives:
- $v_e=\sqrt{\frac{2GM}{R}}$.
- Using $g=\frac{GM}{R^2}$, this can be written as $v_e=\sqrt{2gR}$.
- Escape velocity is independent of the mass and direction of projection of the body — it depends only on the planet.
For the Earth, $v_e\approx11.2\ \text{km/s}$. Notice the neat link to orbital motion: $v_e=\sqrt{2}\times v_o$, where $v_o$ is the orbital velocity close to the surface. This is why a planet with a high escape velocity (like Earth) can retain a thick atmosphere, while the Moon, with $v_e\approx2.4\ \text{km/s}$, has effectively lost its atmosphere.