Potential energy calculator

A potential energy calculator works out the energy an object stores because of its position or shape. Gravitational potential energy is E = mgh, so a 60 kg climber 10 m up stores 60 × 9.81 × 10 = 5,886 J. Elastic potential energy in a stretched spring is E = ½kx². Our Potential Energy Calculator does both, and rearranges each formula for any of its values.

It opens on gravitational energy for a 60 kg mass at 10 m. Switch between Gravitational and Elastic at the top.

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Potential Energy Calculator

Work out gravitational (mgh) and elastic potential energy

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How to use the potential energy calculator

1. Under Calculate, choose Gravitational or Elastic.
2. In the second row, choose what to find. Gravitational has Energy, Mass, Height and Gravity. Elastic has Energy, Extension and Spring constant.
3. Type the values you know. Leave Gravity blank for Earth's 9.81 m/s², or pick a planet from Common values: the Moon, Mercury, Venus, Mars, Jupiter and more.
4. Change units if you need to: mass in grams or tonnes, height or extension in cm or km, energy in kJ.
5. Read the answer and the working. A graph shows the energy against height or extension, with your value marked.

Extra readings come with the answer: for gravitational energy, the speed the object would reach falling that height and the energy per kilogram; for elastic energy, the force on the spring at that extension. Practice and help sets lifting, climbing and spring questions with worked solutions.

The potential energy formulas

Both give the energy in joules (J). Gravitational potential energy grows in step with height: twice as high stores twice the energy. Elastic energy depends on the extension squared, so stretching a spring twice as far stores four times the energy, as long as it stays within its limit of proportionality. Our guides to gravitational potential energy and elastic potential energy cover each formula in depth.

Gravitational potential energy calculator

For gravitational potential energy, choose Gravitational, then what to find: Energy from E = mgh, or the Mass, Height or Gravity. Gravity is Earth's 9.81 m/s² when you leave it blank, and Common values lists the Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus and Neptune, so the same calculator works near the surface of any of them. A 60 kg climber 10 m up stores 5,886 J on Earth, 2,226 J on Mars, where g = 3.71 m/s², and 972 J on the Moon, where g = 1.62 m/s². E = mgh only holds near a surface, where g hardly changes over the height. Our gravitational potential energy guide covers the formula, its derivation and exam use.

Worked examples you can check in the calculator

Example 1: gravitational potential energy

A 60 kg climber climbs 10 m up a wall. How much gravitational potential energy do they gain?

These are the values the calculator opens on. It gives 5,886 J, with 98.1 J per kilogram, and says the climber would reach 14.0 m/s falling the full 10 m with no air resistance.

Example 2: finding the height

A 2.5 kg book gains 49 J when it is lifted onto a shelf. How high is the shelf?

Choose Gravitational, then Height, and type 49 and 2.5. The calculator gives 1.998 m.

Example 3: on the Moon

The same 60 kg climber on a 10 m wall on the Moon, where g = 1.62 m/s², stores only 60 × 1.62 × 10 = 972 J. Pick Moon from Common values and the calculator gives 972 J. Working backwards, the Gravity mode finds g from the energy: 972 ÷ (60 × 10) = 1.62 m/s².

Example 4: elastic potential energy

A spring with spring constant 400 N/m is stretched by 15 cm. Convert to metres first: x = 0.15 m.

Choose Elastic, then Energy, and type 400 and 0.15. The calculator gives 4.5 J, and the force at this extension, F = kx = 60 N.

Here x is the extension in metres and y is the energy in joules. Halving the extension to 0.075 m stores a quarter of the energy, 1.125 J, because of the square.

Example 5: finding the extension

A spring of constant 250 N/m stores 2.0 J. How far is it stretched?

Choose Elastic, then Extension, and the calculator gives 0.1265 m, about 12.6 cm.

Check the pattern:

Potential energy to kinetic energy

When an object falls, its gravitational potential energy turns into kinetic energy. With no air resistance, mgh = ½mv², so v = √(2gh) and the mass cancels. A 0.5 kg ball dropped from 3.2 m has 0.5 × 9.81 × 3.2 = 15.7 J of potential energy, and hits the ground at √(2 × 9.81 × 3.2) = 7.92 m/s. The calculator gives both: 15.696 J, and 7.92364 m/s as the speed if it falls that far. For the kinetic energy side, use the Kinetic Energy Calculator.

What the calculator does not do

• mgh is for heights near a surface. It assumes g stays the same. For satellites and planets, where g changes with distance, the energy is −GMm/r: use the Gravitational Potential Calculator.
• No electric potential energy. The energy between two charges, kQq/r, is not in this calculator.
• Springs within their limit only. ½kx² assumes the spring obeys Hooke's law. Past the limit of proportionality, find the area under the force-extension graph by hand.

Common mistakes

• Using centimetres. Height and extension must be in metres for the answer to be in joules. 15 cm is 0.15 m.
• Forgetting to square the extension. ½ × 400 × 0.15 = 30 is wrong; ½ × 400 × 0.15² = 4.5 J is right.
• Using weight as the mass. If you are given a weight in newtons, that is already mg: E = weight × h.
• Measuring h from the wrong place. h is the change in height, from where the object starts to where it ends.
• Mixing up the two formulas. mgh is for height in a gravitational field; ½kx² is for springs and other stretchy things.

Practise energy in a short timed paper

Potential energy comes up in High School Physics and AP Physics 1 practice, and in GCSE Physics Higher and A-level Physics: Year 1 in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. For springs, the spring constant guide covers Hooke's law.

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