Gravitational potential calculator

A gravitational potential calculator finds the gravitational potential V at a distance r from the centre of a mass M: V = −GM/r, in joules per kilogram. At the Earth's surface it is about −6.26 × 10⁷ J/kg. Our Gravitational Potential Calculator finds V, and the potential energy U = −GMm/r of a second mass m placed at that point, with the working and a graph of potential against distance.

It opens on the potential at the Earth's surface. Choose Potential Energy (U) for the energy of a mass, which opens on a 1,000 kg satellite 400 km up.

Try it

Gravitational Potential Calculator

Calculate gravitational potential energy

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How to use the gravitational potential calculator

1. Choose Potential (V) or Potential Energy (U).
2. Type the mass M as a number times a power of ten, such as 5.972 ×10²⁴ kg for the Earth. For U, also type the second mass m.
3. Type the Radius: the distance from the centre of M, not the height above its surface.
4. Read the answer and the working, and the graph of potential against distance.

Each box takes a number and a power of ten; type 0 in the ×10 box for an ordinary number. Practice and help sets potential questions with worked solutions.

The formulas

The zero of potential is set infinitely far away. A mass near the planet has to be given energy to get out there, so its potential starts below zero. Closer in, the potential is more negative. Our gravitational potential guide explains why, with the potential gradient and equipotentials.

Worked examples you can check in the calculator

Example 1: potential at the Earth's surface

These are the values the calculator opens on, and it gives −6.2563 × 10⁷ J/kg. On the Moon's surface (7.346 × 10²² kg, radius 1.7374 × 10⁶ m) it is −2.82 × 10⁶ J/kg, about 22 times less negative.

Example 2: energy to lift a satellite to orbit height

How much energy does it take to lift a 1,000 kg satellite from the surface to 400 km up? Find U at both distances:

The change is −5.8867 × 10¹⁰ − (−6.2563 × 10¹⁰) = 3.70 × 10⁹ J. Using mgh with g = 9.81 m/s² would give 3.92 × 10⁹ J, too much, because g gets smaller with height. This is why mgh only works near a surface. Moving it into orbit needs more energy again, for its orbital speed.

Example 3: escape

To escape from the Earth's surface, a mass needs enough kinetic energy to raise its potential to zero: ½v² = 6.26 × 10⁷ J/kg, so v = √(2 × 6.2563 × 10⁷) = 11.2 km/s, the escape velocity. The Escape Velocity Calculator works this out directly.

Check yourself:

What the calculator does not do

• One mass at a time. For the potential between the Earth and the Moon, add the two potentials by hand.
• No potential difference mode. For the energy to move between two points, work out U at each and subtract, as in Example 2.
• Outside the mass only. V = −GM/r holds outside a spherical mass, not inside it.

Common mistakes

• Using height instead of radius. r is from the centre: add the planet's radius to any height.
• Dropping the minus sign. Gravitational potential is always negative.
• Using mgh far from the surface. Use the change in U = −GMm/r instead.
• Mixing up V and U. V is per kilogram, in J/kg; U is for a particular mass, in J.

Practise gravitational fields in a short timed paper

Gravitational potential comes up in A-level Physics: Year 2 practice in the UK, and in College Physics (Calculus Based) in the US. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. For the force itself, see our gravitational force calculator guide.

Read next

Physics · Fields and gravityGravitational potential: V = −GM/r explained with examplesGravitational potential explained: the formula V = −GM/r, why it is negative, potential energy, equipotentials and the link to g, with worked examples.