AP® Physics C: Mechanics · Unit 3 · Topic 3.3
Gravitational Potential Energy Far from a Planet
Far from a planet, gravitational potential energy is U = -GMm/r, negative because it's measured relative to zero at infinity. Moving apart raises U toward zero; moving together lowers it further negative.
Near Earth's surface you used U = mgh, a formula built for small heights where gravity barely changes. Once you're talking about satellites, escape speed, or trips between planets, the force itself weakens with distance, so you need the real law: F = GMm/r2. Potential energy is the antiderivative of force, with a sign flip: U(r) = -integral of F dr. Integrating GMm/r2 from infinity in to some radius r gives U = -GMm/r. The constant of integration is chosen so U → 0 as r → infinity. That choice, not the physics, is why the number comes out negative: you're measuring energy relative to a system that has been pulled apart completely.
Negative U just means this configuration has less energy than the zero reference at infinite separation. As two masses move apart, 1/r shrinks, so -GMm/r climbs toward zero — potential energy increases, exactly as it should when you fight gravity to separate things. As they fall together, r shrinks and U drops further negative, releasing energy as kinetic energy. Escape speed comes from demanding total energy be exactly zero at infinity: (1/2)mv2 - GMm/R = 0, giving v_esc = √(2GM/R). This connects directly to orbital mechanics next: a bound orbit has total energy less than zero, trapped in the negative well, while an unbound trajectory has zero or positive total energy and can reach infinity.
This same U also plugs straight into conservation of energy problems: lifting a probe from the surface to some larger radius costs energy equal to the change in U, which is why that calculation looks like GMm times a difference of two reciprocals.
Key terms
- Escape speed
- The minimum speed at a given radius for which total mechanical energy is zero, letting an object reach infinity with zero leftover kinetic energy.
- Zero reference at infinity
- The convention that gravitational potential energy is defined to equal zero when two masses are infinitely far apart.
- Inverse square force
- A force, like gravity, whose magnitude falls off as 1 over the square of the distance between two objects.
- Bound orbit
- An orbit where total mechanical energy is negative, keeping the object trapped and unable to reach infinite separation.
Watch it explained
- How it worksHow does integrating the inverse square force in from infinity produce U = -GMm/r?
- Worked exampleFind the escape speed from a planet with Earth's mass packed into half Earth's radius.
- Worked exampleFind the energy needed to lift a 500 kg probe from Earth's surface out to three Earth radii.
- Common mistakeWhy doesn't a negative potential energy mean the satellite somehow has less than nothing?
Check yourself
End of topic.