Potential Energy Calculator

Free Online Potential Energy Calculator (PE = mgh)

This potential energy calculator instantly solves for gravitational potential energy, mass, or height using the fundamental formula PE = m × g × h — Potential Energy equals mass times gravity times height. Enter any two known values — mass and height to find potential energy, potential energy and height to find mass, or potential energy and mass to find height — in any common unit, and the calculator converts everything automatically, works through a complete step-by-step solution, and draws a labeled diagram showing the object at its height above the ground along with a potential-energy-versus-height chart, so the entire calculation is visible at a glance, not just the final number.

Whether you're a physics student solving a potential energy word problem, an engineer estimating stored energy in a raised load, or just curious how much energy a book on a shelf or a diver on a diving board carries, this calculator handles kilograms, grams, pounds, ounces, and metric tons for mass, meters, centimeters, kilometers, feet, inches, and miles for height, and joules, kilojoules, calories, kilocalories, foot-pounds, and ergs for energy — plus built-in gravity presets for Earth, the Moon, Mars, and Jupiter, or a custom gravity value of your own. It always shows its complete working, step by step.

What Is Gravitational Potential Energy? (Definition and Formula)

Gravitational potential energy is the energy an object has because of its position above a reference point, usually the ground. Any object raised to a height — a book on a shelf, water behind a dam, a diver on a board, or a ball held in the air — has stored energy that can be converted into motion (kinetic energy) if the object is allowed to fall. Potential energy is a scalar quantity, meaning it has magnitude but no direction, and like all forms of mechanical energy, it's measured in joules.

The formula for gravitational potential energy is one of the foundational equations in classical mechanics:

• Potential Energy (PE) = mass (m) × gravity (g) × height (h)
• The SI unit of potential energy is the joule (J), where 1 joule = 1 kilogram-meter²/second² (kg·m²/s²).
• On Earth, gravity (g) is approximately 9.81 m/s².
• Example: a 2 kg book on a shelf 1.5 m above the floor has potential energy = 2 × 9.81 × 1.5 = 29.43 J

Why Height and Gravity Both Matter

Unlike kinetic energy, which depends on velocity squared, potential energy is directly (linearly) proportional to both height and gravity — doubling the height doubles the potential energy, and doubling the gravitational field strength also doubles the potential energy, with no squaring involved. This straight-line relationship is exactly what the PE-vs-height chart on this page shows: as height increases, potential energy increases along a perfectly straight line, whose slope is set by mass × gravity. This is also why the same object at the same height has very different potential energy on different worlds — an astronaut's backpack raised 2 m on the Moon (g ≈ 1.62 m/s²) stores roughly six times less potential energy than the same backpack raised 2 m on Earth (g ≈ 9.81 m/s²), even though the mass and height are identical.

How to Find Mass from Potential Energy and Height

If you already know an object's potential energy and its height above the ground, you can rearrange the formula to solve for its mass.

• Mass (m) = Potential Energy (PE) ÷ (gravity (g) × height (h))
• Example: an object has 294 J of potential energy at a height of 15 m on Earth → Mass = 294 ÷ (9.81 × 15) = 294 ÷ 147.15 = 2 kg
• This rearrangement comes directly from dividing both sides of PE = m × g × h by g × h.

How to Find Height from Potential Energy and Mass

If you know an object's potential energy and its mass, you can solve for its height by rearranging the formula.

• Height (h) = Potential Energy (PE) ÷ (mass (m) × gravity (g))
• Example: a 10 kg object has 500 J of potential energy on Earth → Height = 500 ÷ (10 × 9.81) = 500 ÷ 98.1 ≈ 5.1 m
• This rearrangement comes from dividing both sides of PE = m × g × h by m × g.

The Potential Energy Formula Triangle

Like the work formula, PE = mgh can be visualized with a simple mental shortcut: cover the quantity you're solving for and what remains gives the formula — covering PE leaves m × g × h; covering m leaves PE over (g × h); covering h leaves PE over (m × g). The diagram on this page reinforces this by showing the object at its actual height above the ground, an amber summary box with the resulting potential energy and the exact formula used, and a straight-line chart plotting potential energy against height so the linear relationship is something you can see, not just calculate.

Common Mass, Height, and Energy Units

Mass, height, and potential energy can each be expressed in several different unit systems depending on whether you're working in SI (metric) or imperial units. This calculator supports the most common ones used in physics classes and engineering.

• Mass: kilograms (kg), grams (g), pounds (lb), ounces (oz), metric tons (t)
• Height: meters (m), centimeters (cm), kilometers (km), feet (ft), inches (in), miles (mi)
• Energy: joules (J), kilojoules (kJ), calories (cal), kilocalories (kcal), foot-pounds (ft·lb), ergs
• Gravity: Earth (9.81 m/s²), Moon (1.62 m/s²), Mars (3.71 m/s²), Jupiter (24.79 m/s²), or any custom value

How to Use This Potential Energy Calculator

Choose what you want to find — Potential Energy, Mass, or Height — from the dropdown. Then fill in the two values you already know, picking whichever unit matches your problem, and select which gravity to use — Earth by default, or the Moon, Mars, Jupiter, or a custom value for problems set on other worlds. The calculator instantly converts everything to a common base unit, applies PE = m × g × h (or its rearranged form), and shows the result in your chosen unit — plus automatically converted into joules, kilojoules, and foot-pounds for quick reference.

Scroll down for the complete step-by-step solution, which mirrors exactly how you'd solve it by hand: the formula first, then your real values substituted in with unit conversion shown explicitly, then the arithmetic worked through to the final answer.

Worked Examples

A book on a shelf: 2 kg at 1.5 m on Earth → Potential Energy = 2 × 9.81 × 1.5 = 29.43 J.

A diver on a board: 60 kg at 3 m on Earth → Potential Energy = 60 × 9.81 × 3 = 1765.8 J.

An astronaut on the Moon: 80 kg at 2 m with g = 1.62 m/s² → Potential Energy = 80 × 1.62 × 2 = 259.2 J.

Finding mass: 294 J of potential energy at 15 m on Earth → Mass = 294 ÷ (9.81 × 15) = 2 kg.

Finding height: 500 J of potential energy with a 10 kg mass on Earth → Height = 500 ÷ (10 × 9.81) ≈ 5.1 m.

Potential Energy, Kinetic Energy, and Conservation of Energy

Potential energy is closely linked to kinetic energy through the principle of conservation of mechanical energy: as an object falls, its potential energy converts into kinetic energy, and — ignoring air resistance and friction — the total mechanical energy (PE + KE) stays constant throughout the fall. This is why a ball dropped from a height starts with maximum potential energy and zero kinetic energy, and finishes just before impact with maximum kinetic energy and (relative to the ground) zero potential energy. This relationship is the basis for countless physics problems, from roller coasters and pendulums to dams and hydroelectric power generation.

Real-World Applications of Potential Energy

The potential energy formula appears throughout engineering, construction, and everyday physics:

• Hydroelectric power — water stored behind a dam has enormous gravitational potential energy, which converts to kinetic energy and then electricity as it falls through turbines.
• Construction and cranes — engineers calculate the potential energy of materials lifted to height to size motors, cables, and safety systems correctly.
• Roller coasters — cars are hauled to the top of the first hill to build up potential energy, which then converts into kinetic energy (speed) throughout the ride.
• Elevators and lifts — the potential energy gained by a raised elevator car is central to calculating the motor power and counterweight needed.
• Space and orbital mechanics — the potential energy of a satellite or spacecraft relative to a planet determines the energy needed to change its altitude or escape orbit.

Tips for Solving Potential Energy Word Problems

Most textbook and exam word problems involving potential energy follow a predictable pattern, and a few habits make them far easier to solve correctly:

• Identify which two quantities are given and which one is being asked for — this tells you whether to use PE = mgh, m = PE ÷ (gh), or h = PE ÷ (mg).
• Always convert mass to kilograms and height to meters before plugging into the formula, since the joule is defined in those base units along with meters per second squared for gravity.
• Check whether the problem specifies a planet or moon other than Earth — using 9.81 m/s² by default when a different gravity is given is one of the most common mistakes in these problems.
• Remember that potential energy is always measured relative to a reference height (often the ground, or h = 0) — a stated height should be the distance above that chosen reference point, not an absolute altitude.

Frequently Asked Questions

What is the formula for gravitational potential energy? Potential Energy = mass × gravity × height, where mass is in kilograms, gravity is in meters per second squared, and height is in meters, giving an answer in joules.

How do you calculate potential energy in joules? Multiply the mass in kilograms by the gravitational acceleration in m/s² and by the height in meters. For example, a 5 kg object at 4 m on Earth has potential energy = 5 × 9.81 × 4 = 196.2 J.

What is the difference between potential energy and kinetic energy? Potential energy is stored energy based on position (such as height), while kinetic energy is the energy of motion. Together they make up an object's total mechanical energy.

Can this calculator solve for mass or height, not just potential energy? Yes — switch the 'What do you want to find?' dropdown to Mass or Height, and the calculator rearranges the formula automatically and solves for whichever value you need.

What value of gravity should I use? Use 9.81 m/s² for Earth, which is the default. This calculator also includes presets for the Moon, Mars, and Jupiter, plus a custom gravity option for any other value.

What is the SI unit of potential energy? The SI unit of potential energy is the joule (J), the same unit used for kinetic energy, work, and all other forms of energy.