Potential Energy Calculator

This potential energy calculator allows you to determine the energy stored in an object due to its position or deformation. Select gravitational potential energy (PE = mgh) for an object raised to a certain height, and elastic potential energy (PE = 1/2*k*x²) for a stretched or compressed spring. By entering known values and leaving one field blank, that field can be calculated. You can therefore determine energy, mass, height, gravity, spring constant, or displacement on the same screen.

What is potential energy?

Potential energy is the stored energy an object has because of its position or state and can be released and turned into motion at any time. It's not a property that belongs to just one object but rather a system property which is always measured relative to a chosen zero reference level. When you place a book on a shelf, you store potential energy in the book due to gravity. If you stretch a rubber band, you store elastic energy in it. When you move an object, the stored energy is converted into kinetic energy or motion energy.

There are several forms of potential energy. The gravitational and elastic potential energies dealt with in this tool are the first two types learned in physics. In addition there is electrical potential energy stored between charged particles, chemical energy stored in the bonds of fuels or food, and nuclear energy bound within the nucleus of atoms.

Potential energy due to gravity

Gravitational potential energy is the energy an object possesses because of its position relative to a gravitational field. It equals the product of the mass of the object, the acceleration due to gravity, and the height above some reference point.

Here m is the mass in kilograms, g is the acceleration due to gravity on Earth in meters per second squared and h is the height in meters. The unit of energy is Joule. On Earth, g is about 9.81 meters per second squared. For example, if you raise a body with a mass of 3 kg to a height of 5 m, then the stored energy will be:

The calculator includes Earth's gravity by default but can be changed to account for the Moon, Mars or other celestial bodies. It can also be used to calculate gravity from known energy by filling in appropriate fields.

Calculation of mass, height and Earth's gravity:

Since PE = m g h is an equation with four variables, it can be converted into a form to solve for any variable. When you enter the field that you want to find the value of, the calculator will automatically use the correct formula.

Potential energy (elastic energy)

The potential energy is the energy stored by stretching or compressing a spring or an elastic body. It depends on the spring constant, which measures stiffness, and the displacement from rest length.

where k is the spring constant in newtons per meter and x is the displacement in meters. Because the displacement is squared, it doesn't matter whether the spring stretches or compresses. The same energy will be stored for a given stretch. If k = 200 newtons per meter and the spring is stretched by 0.5 meters, then the stored energy is:

If the spring is stretched twice as far, the stored energy quadruples. This is because the formula contains a squared term. The calculator also shows the spring force F = k x which is the force you feel when trying to hold the spring at this position.

Zero level

Potential energy has no absolute value and is always measured relative to a horizontal plane chosen as the zero level. If a book is on a table, its height can be measured from either the table or the floor, leading to two different values. Only the change in potential energy between two positions has any physical significance, and this change is independent of which zero level is chosen. Choose the zero level that makes the problem easiest. Usually this will be the lowest point the object can reach.

Can potential energy be negative?

Yes. If the object is below your chosen zero level then its gravitational potential energy will be negative. This is because you would have to do work on it to raise it up to the zero level. If you measure height from ground level, a ball in a well has negative potential energy. The sign just indicates which side of the zero level the object is on. In physics only differences are important so negative values aren't really an issue.

Is the path important?

Gravity is a conservative force. This means that the potential energy due to gravity depends only on height and not on the path taken to get there. Whether you carry a suitcase straight up a ladder or push it up a long hill, the resulting potential energy is exactly the same as long as the final height is the same. On an incline, the same amount of work is done with less force over a longer distance, but the stored energy at the top doesn't change.

The gravity of other celestial bodies

Since potential energy is proportional to gravity, the amount of energy stored at a given height can vary widely between different celestial bodies. If an object were lifted one meter, it would store about six times less energy on the Moon than on Earth, while much more energy would be stored near the Sun. A table comparing celestial bodies shows how much energy an object would store on each planet.

Potential and kinetic energy

Energy is never lost but only changes forms. When an object falls, the potential energy stored by gravity is converted into kinetic energy causing its speed to increase. As an object rises, the direction of this exchange reverses and its speed decreases. Assuming that potential energy equals kinetic energy, then the velocity of an object falling from a height h when it hits the ground is the square root of 2 g h, which is independent of the mass of the object. A calculator gives impact velocity and fall time, and an energy diagram shows how potential and kinetic energy are constantly exchanged during the fall.

A typical example is roller coasters. They store potential energy by climbing the first hill, convert that to kinetic energy as they descend, and then store potential energy again as they climb the next hill. The same applies to swings: at the lowest point height is converted into speed while at the highest point speed is converted into height. In hydroelectric power plants, the potential energy of water stored behind a dam at high elevation is converted into kinetic energy of falling water and then into electrical energy.

Example calculation:

How much potential energy does a book with mass of 2 kilograms have if it is on a shelf at height of 10 meters? If it falls down, how high will its speed be when it hits the ground? First we calculate the potential energy.

When the book falls, all of its potential energy is converted into kinetic energy. So when it hits the ground, its speed will be the square root of 2 times 9.81, multiplied by 10, which is about 14 meters per second. It takes about 1.4 seconds for the book to hit the ground. The book was only a short distance off the ground, but it stored nearly 200 joules of energy. This is the amount of energy that gravity exerted on the book as it lifted it up.

Why are joules used?

The unit for potential energy is the Joule. This is the SI unit of energy and is also used for kinetic energy, work, and heat. As they are both measures of the same physical quantity, they can be freely converted into each other. One joule is roughly equivalent to the amount of energy it takes to lift a small apple (about 100 grams) one meter high. Depending on whether your answer will be large or small, the calculator also has kilojoules, megajoules, kilowatt hours, food calories and foot-pounds available so you can read off the result in whatever unit is most convenient for the problem at hand.

Common mistakes:

A few small oversights can easily lead to mistakes. The first mistake is measuring height from the wrong reference surface or confusing height with total distance traveled. Only vertical height above the zero reference plane is included in the calculation. Another mistake is forgetting to square the displacement when dealing with elasticity. Energy is proportional to the square of x, not just proportional to x. A third mistake is mixing up units. To get Joules directly, kilograms, meters and meters per second squared must be used. However, this calculator converts the input units so using grams or centimeters is no problem at all.