Gravitational Potential Energy
Energy of a mass in another body's gravitational field.
Formula
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Example
It is negative because energy is zero at infinite separation
How it works
This formula gives the gravitational potential energy of two masses separated by a distance r, measured relative to zero energy at infinite separation. The negative sign appears because gravity is an attractive force: as two objects move closer together, the system loses potential energy and that energy can be released as kinetic energy or other forms. It is derived by integrating the gravitational force GMm/r^2 from infinity down to r.
When to use it
Use this when you need the energy associated with an object in a gravitational field, especially in orbital mechanics, escape velocity, satellite motion, and astrophysics. It is most useful when the separation between two bodies changes and you want the energy difference between positions.
Worked example, step by step
A 1200 kg satellite orbits Earth at a distance of 7.0 × 10^6 m from Earth’s center. Use G = 6.67 × 10^-11 N m^2 kg^-2 and M_Earth = 5.97 × 10^24 kg. U = -GMm/r U = -(6.67 × 10^-11)(5.97 × 10^24)(1200) / (7.0 × 10^6) U = -6.82 × 10^10 J Final answer: U ≈ -6.8 × 10^10 joules
Common mistakes to avoid
- Forgetting the minus sign and reporting gravitational potential energy as positive
- Using the distance from a planet's surface instead of the distance from its center
- Mixing up this formula with near-surface energy mgh, which only works when the height change is small compared with the planet's radius
Did You Know?
The energy is negative because gravity is attractive — you must add energy to pull two bodies apart to infinity.
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