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Gravitational Potential Calculator

Find the gravitational potential near a mass.

How it is calculated

The tool evaluates V = −G × M ÷ r, with G = 6.674 × 10−11 N·m²/kg², M the mass of the attracting body in kilograms and r the distance from its center in meters. The answer is in joules per kilogram (J/kg) and is always negative, because zero potential is defined at infinite distance. Both mass and distance must be greater than zero, otherwise no result appears. Very large or small numbers can be typed in scientific notation, such as 5.972e24. The formula treats the body as a point mass or a uniform sphere and is valid only outside it; it does not compute the potential inside a planet, and it does not add contributions from other bodies such as the Sun or Moon.

Worked example

Take the Earth's mass, 5.972 × 1024 kg, and a distance of 6.371 × 106 m, roughly the radius at the surface. First G × M = 6.674 × 10−11 × 5.972 × 1024, about 3.9857 × 1014. Dividing by r gives 3.9857 × 1014 ÷ 6.371 × 106, about 62,560,238.58. With the minus sign, the tool shows -62,560,238.58 J/kg, which is the energy per kilogram needed to reach infinity from there.

Try this example →

Common mistakes

  • Entering the altitude above the surface as the radius. The distance must be measured from the center of the body, so add the planet's radius to the altitude.
  • Entering kilometers. The radius is in meters; 6,371 km must be typed as 6371000 or 6.371e6.
  • Expecting a positive result. The negative sign is correct, and a value closer to zero means a point farther from the mass.
  • Confusing potential with potential energy. The result is per kilogram; multiply by the object's mass to get joules.
  • Using the formula inside a body, such as at a depth in the Earth. It applies only outside the mass.

About this calculator

Gravitational potential is defined as V = −G·M/r, the potential energy per unit mass at a distance r from a mass M, where G is the gravitational constant. It is always negative, reflecting that gravity is attractive and that a mass at infinite distance is taken as the zero-energy reference point; potential becomes more negative closer to the mass, representing a deeper energy well. Escape velocity, v_esc = √(2GM/r), is derived directly from this potential by setting kinetic energy equal to the energy needed to reach zero potential (infinity).

Astrophysicists and orbital-mechanics engineers use gravitational potential to compute orbital energy, design spacecraft trajectories, and calculate escape velocities from planets or moons, while physics students use it to connect gravitational force to potential energy through the fundamental relationship F = −dV/dr (per unit mass). Planetary scientists also use it to compare the “depth” of different bodies’ gravity wells when planning missions or studying tidal effects.

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