Gravitational Potential Energy Calculator

How to use this calculator

This calculator finds the gravitational potential energy stored in an object at height h above a reference point. Three inputs are required.

Mass (kg): The mass of the object in kilograms. Enter any positive value. A person weighing 70 kg, a car at 1200 kg, or a water tank at 500 kg all work fine.

Height (m): The vertical distance above the chosen reference point (usually the ground). For an object on a shelf 1.5 m above the floor, enter 1.5.

Gravitational acceleration (m/s²): Set automatically by the planet preset buttons (Earth = 9.81, Moon = 1.62, Mars = 3.72, Jupiter = 24.79, Venus = 8.87, Saturn = 10.44). For a custom environment, select Custom and type any value.

Click Calculate to see GPE in joules (J), the object’s weight in newtons (N), GPE in kilojoules (kJ), GPE in kilowatt-hours (kWh), and a chart showing how GPE grows linearly with height.

What is gravitational potential energy?

Gravitational potential energy (GPE) is the energy stored in an object because of its position in a gravitational field. More precisely, it is the work that would be done by gravity on the object as it falls from its current height to the reference level. The higher the object and the greater its mass, the more energy is stored.

GPE is a form of potential energy: energy that is not currently causing motion but has the capacity to do so. A book on a shelf, a diver on a high board, water behind a dam, and a satellite in orbit all possess gravitational potential energy relative to some reference point.

Where:

• m = mass in kilograms (kg)
• g = gravitational acceleration in m/s² (9.81 m/s² on Earth’s surface)
• h = height above the reference point in meters (m)
• GPE = gravitational potential energy in joules (J)

The reference point concept

One of the most important ideas in GPE calculations: the value of GPE depends entirely on the chosen reference point. GPE is always a relative quantity.

If you place a book 2 m above the floor and choose the floor as h = 0, the GPE is mgh calculated at h = 2 m. If you instead choose the table surface (1 m above the floor) as the reference, the same book has GPE calculated at h = 1 m. Neither answer is wrong: they are answers to different questions.

In practical problems, you choose the reference level for convenience. Common choices:

• Ground level: Most useful for falling objects, projectiles, and everyday scenarios.
• Lowest point of the system: Useful for pendulums, roller coasters, and oscillating systems.
• Infinity: Used in orbital mechanics and gravitational field theory. At infinity, GPE = 0, so all bound objects have negative GPE.

For day-to-day calculations, the absolute value of GPE rarely matters. What matters is the change in GPE (ΔGPE), because that change converts to or from kinetic energy or heat.

Derivation from work done against gravity

The formula GPE = mgh comes directly from the definition of work. Work done by a force is W = F × d × cos(θ), where θ is the angle between force and displacement.

To lift an object of mass m from height 0 to height h at constant velocity, you must apply a force equal to the object’s weight, F = mg, upward, while the object moves distance h upward. The angle between force and displacement is 0°, so:

W = mg × h × cos(0°) = mgh

This work you do against gravity is stored as gravitational potential energy. When you release the object, gravity does the same amount of work back on it (positive work now, since gravity points down and the object moves down), converting GPE back to kinetic energy.

Energy conservation: GPE and kinetic energy

In a system without friction or air resistance, total mechanical energy is conserved:

As an object falls from height h, it loses GPE and gains an equal amount of kinetic energy. Just before hitting the ground:

• GPE lost = mgh
• KE gained = ½mv²
• Setting them equal: mgh = ½mv², so v = √(2gh)

This is why a ball dropped from 5 m hits the ground at v = √(2 × 9.81 × 5) ≈ 9.9 m/s, regardless of its mass. The mass cancels out of the equation.

The work-energy theorem gives us a more general version: the net work done on an object equals its change in kinetic energy. Gravity does positive work as an object falls, increasing KE; an applied force does positive work when lifting an object, increasing GPE.

GPE in hydroelectric power

Hydroelectric dams are one of the largest practical applications of gravitational potential energy. Water stored in a reservoir at height h has GPE = mgh relative to the turbines below. As water flows down through penstocks (large pipes), GPE converts to KE, which then drives turbines to generate electricity.

The power available from a dam is:

P = ρ × Q × g × h × η

Where:

• ρ = density of water (1000 kg/m³)
• Q = flow rate in m³/s
• g = 9.81 m/s²
• h = head (height difference) in meters
• η = efficiency of the turbine-generator system (typically 0.85 to 0.95)

The Three Gorges Dam in China has a head of about 80 m and a flow rate capacity of roughly 30,000 m³/s, giving a theoretical power of 1000 × 30,000 × 9.81 × 80 × 0.9 ≈ 21,000 MW. This is why large dams with high reservoirs are such powerful energy sources: GPE scales with both mass (water volume × density) and height.

GPE in pendulums and roller coasters

A pendulum constantly converts between GPE and KE as it swings. At the highest points of the swing, velocity is zero and all mechanical energy is stored as GPE relative to the bottom. At the lowest point, all GPE has converted to KE and velocity is at its maximum.

For a pendulum bob of mass m swinging to a maximum height h above the rest position:

• GPE at top = mgh
• KE at bottom = ½mv²_max = mgh, so v_max = √(2gh)

The period of the pendulum (T = 2π√(L/g)) depends on length L and gravity g, not on mass or amplitude (for small angles). This is the isochronous property that made pendulum clocks accurate timekeepers.

Roller coasters work on the same principle. The first hill (the tallest) is always given the most height by the motor chain lift. From then on, energy conservation (with friction losses) governs the ride. Each subsequent element must be at a lower elevation than the previous one, or the coaster would need additional energy input.

GPE on other planets

Since GPE = mgh, changing the gravitational acceleration g directly scales GPE. The same object at the same height stores different amounts of energy on different worlds.

Body	g (m/s²)	GPE for 10 kg at 5 m
Earth	9.81	490.5 J
Moon	1.62	81.0 J
Mars	3.72	186.0 J
Jupiter	24.79	1,239.5 J
Venus	8.87	443.5 J
Saturn	10.44	522.0 J

This matters for mission planning. A rover on Mars requires much less energy to lift equipment to a given height. Astronauts on the Moon can jump far higher with the same muscular effort, because less GPE is required per meter of elevation.

For spacecraft, GPE relative to a planet is usually defined with the reference at infinity (GPE = 0 at infinite distance). In that convention, all bound orbits have negative GPE, and the total mechanical energy (KE + GPE) of an object in orbit is also negative. Escape velocity is the speed at which total mechanical energy reaches zero.

GPE in satellites and orbital mechanics

For orbital mechanics, GPE takes a different form. Near Earth’s surface, g is approximately constant, and GPE = mgh is a good approximation. But for orbits at significant altitude, gravity weakens with distance, and the correct formula is:

Where G is the gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), M is the mass of the planet, m is the satellite’s mass, and r is the distance from the planet’s center. The negative sign reflects the convention that GPE = 0 at infinite distance.

At low Earth orbit (altitude ~400 km, r ≈ 6.78 × 10⁶ m), the International Space Station has an orbital speed of about 7.66 km/s. Its total mechanical energy is negative (bound orbit). To escape Earth’s gravity entirely from this orbit, it would need to increase its speed to about 10.8 km/s, raising total energy to zero or above.

Tidal energy is also GPE in a broader sense: the Moon’s gravity raises tidal bulges in Earth’s oceans, storing gravitational potential energy. As tides ebb and flow, this energy dissipates as heat in shallow seas and seafloor friction.

Molecular and atomic scale GPE

At the molecular scale, intermolecular forces create potential energy wells similar in concept to gravitational potential energy. When two molecules attract each other (as in van der Waals forces), bringing them together from large separation releases energy. The potential energy decreases (becomes more negative) as separation decreases, reaching a minimum at the equilibrium bond distance.

Chemical bonds store energy in this way. Breaking a chemical bond requires adding energy to overcome the potential well; forming a bond releases energy. Exothermic reactions release more energy from bond formation than they consume in bond breaking, resulting in a net GPE decrease (measured as enthalpy change).

This is a completely different physical mechanism from gravitational PE, but the mathematical framework (potential energy as a function of position) is the same. Physicists and chemists use the concept of potential energy wells and barriers extensively in quantum mechanics to describe tunneling, reaction rates, and energy levels.

Tidal and geothermal energy connections

Tidal energy is essentially gravitational potential energy stored in the movement of ocean water relative to the Earth-Moon-Sun system. The Moon’s gravity raises water by roughly 0.5 to 1 m in open ocean (more in coastal areas with resonant geometries, such as the Bay of Fundy where tides reach 16 m). Tidal barrages capture this elevation difference to generate electricity, much like run-of-river hydropower.

Geothermal energy, while primarily thermal, also involves gravitational effects. Magma rises through the crust partly due to its lower density compared to surrounding rock (buoyancy, which is itself a gravitational effect). The heat in Earth’s interior was partly deposited by gravitational compression during planetary formation (converting GPE to heat as material fell inward and compressed), and partly from radioactive decay.

These connections illustrate that GPE is not just a textbook abstraction: it underlies large-scale energy systems that power human civilization and shape planetary geology.

Worked examples

Example 1: Water in a tank

A storage tank holds 2000 kg of water at an average height of 15 m above the turbine.

GPE = 2000 × 9.81 × 15 = 294,300 J = 294.3 kJ

If the turbine converts this at 90% efficiency: Electrical energy = 0.9 × 294.3 = 264.9 kJ

Example 2: Comparing two planets

A 5 kg rock at 10 m height. Compare Earth vs Mars.

Earth: GPE = 5 × 9.81 × 10 = 490.5 J

Mars: GPE = 5 × 3.72 × 10 = 186 J

Mars GPE is 37.9% of Earth GPE.

Example 3: Speed at impact

A 1 kg ball is dropped from 20 m (no air resistance).

GPE at top = 1 × 9.81 × 20 = 196.2 J

KE at bottom = 196.2 J = ½ × 1 × v²

v = √(2 × 196.2) = √392.4 ≈ 19.8 m/s ≈ 71.3 km/h