Potential Energy Calculator

What Is the Gravitational Potential Energy Formula?

The gravitational potential energy (PE) of an object is the energy stored due to its position in a gravitational field. The classic formula is:

• Formula: PE = m × g × h
• PE: Potential energy (joules, J)
• m: Mass (kilograms, kg)
• g: Acceleration due to gravity (m/s², Earth default: 9.81)
• h: Height above reference (meters, m)

Common Applications

Potential Energy as “Stored Work”

While we often think of potential energy as “energy of position,” it’s more deeply understood as stored work. The gravitational potential energy of an object is precisely equal to the amount of work required to lift it against the force of gravity to its current height.

Recall that work is defined as force multiplied by distance (Work = Force × Distance). To lift an object of mass (m), you must apply an upward force equal to its weight, which is the force of gravity acting on it (F = m × g). If you lift it a vertical distance (h), the work you have done is:

Work = (m × g) × h

This is exactly the formula for potential energy. The energy you expended to lift the object isn’t lost; it’s stored in the object as potential energy. This stored energy can be converted back into other forms, such as kinetic energy (the energy of motion), if the object is allowed to fall. Our Potential Energy Calculator helps quantify this stored work for any given scenario.

The Crucial Role of the Reference Point (h=0)

One of the most important concepts in understanding potential energy is that it is relative. The value you calculate depends entirely on where you define the “zero height” level, or the reference point.

There is no universal, absolute zero for height. You must choose a convenient reference point for your calculation.

• If you lift a book from the floor to a table, you can set the floor as h=0. The book’s potential energy is then calculated relative to the floor.
• If you then lift the same book from the table to a high shelf, you could set the table as h=0. The change in potential energy would be calculated based on the height difference from the table to the shelf.

Can Potential Energy Be Negative?

Yes, potential energy can be negative. This simply means the object is located below your chosen reference point. For example, if you stand on a bridge and define the bridge’s surface as h=0, a boat passing underneath you would have a negative height and therefore negative gravitational potential energy relative to you.

What truly matters in physics problems is the change in potential energy (ΔPE), which is independent of your chosen reference point. The difference in PE between the table and the shelf will be the same whether you set your h=0 at the floor, the table, or the center of the Earth.

The Interplay Between Potential and Kinetic Energy

Potential energy (PE) and kinetic energy (KE) are the two main components of an object’s mechanical energy. They are intrinsically linked by one of the most fundamental principles in physics: the Law of Conservation of Energy.

This law states that in an isolated system where there are no non-conservative forces like friction or air resistance, the total mechanical energy remains constant. Energy is not created or destroyed; it simply transforms between potential and kinetic forms.

Total Energy = PE + KE = Constant

The Roller Coaster Example:

A roller coaster is the perfect illustration of this energy conversion:

1. At the Top of the Lift Hill: The coaster car is pulled slowly to its maximum height. Here, its velocity is near zero (minimal KE), but its height is at a maximum, so its potential energy is at its peak.
2. Going Down the First Drop: As the coaster descends, its height (h) decreases rapidly. This loss in potential energy is converted directly into kinetic energy. The car’s velocity (v) increases, and it reaches its maximum kinetic energy at the very bottom of the hill (where h is at its minimum).
3. Going Up the Next Hill: As the coaster climbs again, it uses its kinetic energy to do work against gravity. Its velocity decreases, and the kinetic energy is converted back into potential energy.

This continuous transformation from stored energy (PE) to energy of motion (KE) and back again is what powers the entire ride. Our Potential Energy Calculator can determine the stored energy at any high point on the track.

Beyond Gravity: Other Types of Potential Energy

While this calculator focuses specifically on gravitational potential energy, it’s important to recognize that “potential energy” is a broad concept that applies to any system with a conservative force. Here are a few other common types:

Elastic Potential Energy

This is the energy stored in an elastic object as a result of its deformation, such as stretching a rubber band or compressing a spring. The formula for the potential energy stored in a spring is:

PE_elastic = ½kx²

Where k is the spring constant (a measure of its stiffness) and x is the distance the spring is compressed or stretched from its equilibrium position. This is the energy that powers everything from a pogo stick to a car’s suspension system.

Chemical Potential Energy

This is the energy stored in the chemical bonds of a substance. It is released during a chemical reaction. For example, the food we eat contains chemical potential energy that our bodies convert to do work. The fuel in a car and the charge in a battery are also examples of stored chemical potential energy.

Potential Energy on Other Worlds

The force of gravity is not the same everywhere. An object’s weight—and therefore its potential energy at a given height—changes depending on the celestial body it’s on. Our calculator allows you to adjust the value of ‘g’ to find the potential energy on other planets or moons.

Here are the approximate surface gravity values for various bodies in our solar system: