Derive and apply the concept of gravitational potential energy in a radial gravitational field, U = -G M m / r, including the concept of escape velocity

What this dot point is asking

NESA wants you to extend the idea of gravitational potential energy from the near-Earth $U = mgh$ approximation to the full radial-field formula $U = -G M m / r$, explain why it is negative, and apply it to problems including escape velocity. This dot point underpins every energy-based question on orbital motion in the second half of Module 5.

The answer

From $mgh$ to the radial formula

Near Earth's surface, gravitational field strength $g$ is approximately constant, and $U = mgh$ works fine. At astronomical scales, $g$ falls off as $1/r^2$, so the potential energy must be obtained by integrating the gravitational force from infinity inward:

The negative sign reflects two choices:

1. Zero potential energy at infinity (the natural reference for radial fields).
2. Attractive force, so moving inward releases energy.

A bound mass (closer than infinity) therefore has $U < 0$.

Why negative U makes physical sense

Imagine releasing a stationary object from far away. Gravity does positive work pulling it inward, increasing kinetic energy. By conservation of energy, potential energy must decrease. Since we set $U = 0$ at infinity, $U$ becomes negative as the object approaches the source.

The deeper into the well, the more negative $U$ becomes. To free a mass from the well (push it to infinity), positive work must be done equal to $|U| = \frac{G M m}{r}$.

Escape velocity

Escape velocity $v_{\text{esc}}$ is the minimum speed needed at a distance $r$ for an object to reach infinity with zero remaining kinetic energy. By conservation of energy:

Solving:

Key features:

• Independent of the mass of the escaping object.
• Depends on the mass $M$ of the source and the launch distance $r$.
• At Earth's surface: $v_{\text{esc}} \approx 11.2$ km/s.
• For a black hole, $r$ shrinks until $v_{\text{esc}}$ approaches the speed of light.

Change in potential energy

When a mass moves from $r_1$ to $r_2$:

If $r_2 > r_1$ (moving away), $\Delta U > 0$: potential energy increases (becomes less negative).

Worked example with numbers

Find the work needed to lift a $500$ kg satellite from Earth's surface ($r_1 = R_E = 6.37 \times 10^6$ m) to an altitude of $1000$ km ($r_2 = 7.37 \times 10^6$ m). Take $M_E = 5.97 \times 10^{24}$ kg.

$\Delta U = G M_E m \left(\frac{1}{r_1} - \frac{1}{r_2}\right)$

$\Delta U = 6.67 \times 10^{-11} \times 5.97 \times 10^{24} \times 500 \times \left(\frac{1}{6.37 \times 10^6} - \frac{1}{7.37 \times 10^6}\right)$

$\Delta U = 1.99 \times 10^{17} \times \left(1.570 \times 10^{-7} - 1.357 \times 10^{-7}\right)$

$\Delta U = 1.99 \times 10^{17} \times 2.13 \times 10^{-8} = 4.24 \times 10^9$ J.

About $4.2$ GJ of work is required, ignoring kinetic energy.

Try it: Escape velocity calculator - choose Earth, Moon, Mars, Jupiter or Sun (or enter custom values) and get $v_{\text{esc}}$ at the surface or any altitude.

Common traps

Forgetting the negative sign. $U = -G M m / r$, not $+G M m / r$. The sign matters for energy conservation.

Using $m g h$ for astronomical distances. This approximation only works for small altitude changes near a planet's surface where $g$ is roughly constant. For satellites and planetary orbits, use the radial formula.

Wrong reference point. Zero potential energy is at infinity, not at the planet's surface or centre.

Confusing escape velocity with orbital velocity. Orbital velocity at radius $r$ is $v_{\text{orb}} = \sqrt{G M / r}$. Escape velocity is $\sqrt{2}$ times larger: $v_{\text{esc}} = \sqrt{2} \cdot v_{\text{orb}}$.

Assuming escape velocity depends on the object's mass. It does not. A pebble and a rocket need the same escape velocity (though the rocket needs more total energy because $E = \frac{1}{2} m v^2$).

In one sentence

Gravitational potential energy in a radial field is $U = -G M m / r$ (with zero at infinity), and the escape velocity is the speed $v_{\text{esc}} = \sqrt{2 G M / r}$ at which an object's total energy is zero and it just escapes the source's gravitational well.