Potential Energy Calculator

What is gravitational potential energy?

Potential energy is energy stored in a body because of where it is. The gravitational kind depends on height: lift a mass against gravity and you do work on it, and that work stays stored until the mass is allowed to fall again. A brick on a scaffold, water behind a dam and a roller coaster at the top of its first hill all hold energy of this sort.

The stored amount depends on three things: the mass, the height above a chosen zero level, and the strength of gravity. The zero level is a free choice. Only the difference in potential energy between two heights has physical meaning, so you can measure from the floor, the table top or sea level, whichever suits the problem.

How to use the calculator

• Potential energy: enter the mass and the height. The result appears in joules (kilojoules or megajoules when large), watt-hours, calories and foot-pounds.
• Mass: enter the energy and the height. The result is a mass in kilograms and pounds, or in grams, tonnes and tons when it is very small or very large.
• Height: enter the energy and the mass. The result is shown in metric and US length units, and the units chosen grow with the size of the answer, from millimetres for tiny heights to kilometres for huge ones.

The gravity field starts at 9.80665 m/s², the standard acceleration at the Earth's surface. You can switch its unit to g (where 1 g is that same value) or type any other number, for example 1.62 m/s² for the Moon. Each input has its own unit menu, so you may mix pounds, feet and kilograms freely.

The potential energy formula

PE = m × g × hm in kilograms, g in m/s², h in metres, PE in joules

The other two modes simply rearrange it:

m = PE ÷ (g × h)

h = PE ÷ (m × g)

This form assumes gravity is the same at every height involved, which is an excellent approximation for anything in everyday reach of the ground. For very large distances, such as satellites, the field weakens with distance and the energy is −G·M·m ÷ r instead, where r is measured from the centre of the planet.

Worked examples

A book on a shelf. A 2.5 kg book sits 1.8 m above the floor. PE = 2.5 × 9.80665 × 1.8 ≈ 44.13 J, which is about 0.0123 Wh. Dropping it turns exactly that much energy into motion and then into heat and sound.

Finding the mass. Raising an object 2 m stores 500 J. Then m = 500 ÷ (9.80665 × 2) ≈ 25.49 kg.

How high can a kilowatt-hour lift? One kWh is 3,600,000 J. Divide by the weight force on a 75 kg person (75 × 9.80665 ≈ 735.5 N) and you get h ≈ 4,895 m. In theory, one kilowatt-hour of electricity would lift that person almost 4.9 km, a reminder of how small a unit of mechanical work is next to household energy bills.

The same lift on other worlds. Only g changes. For 10 kg raised 5 m:

The gravity values are rounded figures; published numbers vary slightly with the definition used.

From height to speed

When an object falls freely, potential energy turns into kinetic energy and, ignoring air resistance, PE lost equals KE gained. Setting m·g·h = ½·m·v² and cancelling the mass leaves a result that does not depend on the mass at all:

v = √(2 × g × h)

A stone dropped from 20 m reaches √(2 × 9.80665 × 20) ≈ 19.8 m/s, about 71 km/h, just before it lands. To explore the motion side, use the kinetic energy calculator, and to find how fast speed builds up use the acceleration calculator. Energy stored in a stretched or compressed spring is a different mechanism, covered by the elastic potential energy calculator.

Common mistakes

• Measuring height from the wrong place. The height is relative to your chosen zero level, so state it. Moving the zero shifts every answer by the same amount.
• Mixing mass and weight. Enter kilograms or pounds as mass. If you only know a weight in newtons, the mass is the weight divided by g.
• Mixing up ton and tonne. The mass menu has a metric tonne (1,000 kg) and a US short ton (2,000 lb, about 907 kg).
• Using the formula at huge heights. Beyond a few hundred kilometres the pull of gravity falls noticeably, and constant g overstates the energy.

Related tools are collected on the physics calculators page.