Gravitational Potential Energy Calculator

Gravitational potential energy comes in two forms:

Near Earth’s surface (simple approximation): U = mgh

General formula (exact, valid anywhere): U = −GMm/r

Where:

• U = Gravitational potential energy (joules)
• m = Mass of the smaller object (kg)
• g = Local gravitational acceleration (9.81 m/s² on Earth’s surface)
• h = Height above reference point (m)
• G = Gravitational constant = 6.674 × 10⁻¹¹ N·m²/kg²
• M = Mass of the central body
• r = Distance between centers (m)

Why is U = −GMm/r negative? The convention is that U = 0 at r → ∞ (infinitely far apart). Since gravity is attractive, you must add energy to separate masses, so bound systems have negative potential energy.

The negative sign is not a bookkeeping quirk. It is what “bound” means. A satellite with total energy below zero cannot escape no matter which direction it points; one with total energy above zero is on an unbound hyperbolic path and will never come back. The dividing line at exactly zero is the parabolic escape trajectory, which is where the escape-velocity formula comes from.

Escape velocity: The escape velocity from distance r is: v_esc = √(2GM/r) This is derived by setting kinetic energy equal to the magnitude of potential energy: ½mv² = GMm/r

Which formula should you use?

Use mgh whenever the height is small compared with the radius of the planet, which covers essentially every everyday problem: dropping a hammer, lifting a box, a roller coaster, a hydroelectric dam. The error is about 0.03% per kilometre of altitude on Earth, so even a skyscraper is well inside the approximation.

Switch to −GMm/r the moment altitude becomes a serious fraction of the radius. Satellites, launch trajectories and anything leaving the atmosphere need the exact form, because mgh keeps charging you the same energy per metre all the way to infinity and would predict that escape is impossible.

Worked example, lifting a person up a flight of stairs

A 70 kg person climbs 3.5 m of stairs. U = mgh = 70 × 9.81 × 3.5 = 2,403 J. That is about 0.57 food Calories, which is the sort of arithmetic that explains why stair-climbing burns less than people hope: your muscles are only about 25% efficient, so the body spends roughly 2.3 Calories to deliver those 0.57.

Worked example, the International Space Station

The ISS orbits at about 420 km altitude, so r = 6,371 + 420 = 6,791 km. For a 1 kg mass:

U = −(6.674 × 10⁻¹¹ × 5.972 × 10²⁴ × 1) / 6.791 × 10⁶ = −5.87 × 10⁷ J

At the surface it would be −6.26 × 10⁷ J, so getting that kilogram up there costs about 3.9 MJ of potential energy alone. The kinetic energy needed to stay in orbit is roughly 30 MJ on top, which is why reaching orbit is overwhelmingly about going sideways fast rather than going up.

Body mass reference: