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Potential Energy Calculator

Potential energy equals mass times gravity times height

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Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object because of its height above a reference point. The higher or heavier the object, the more potential energy it has. When released, this energy converts to kinetic energy as the object falls.

E = m × g × h

Solve for Mass

Rearranging the potential energy equation to find the mass of an object, given its potential energy, the acceleration due to gravity, and its height above the reference point.

m = E / (g × h)

Solve for Gravity

Determine the gravitational acceleration at a given location from the potential energy, mass, and height. Useful for comparing gravity on different planets or at different altitudes.

g = E / (m × h)

Solve for Height

Find the height of an object above the reference point, given its potential energy, mass, and gravitational acceleration.

h = E / (m × g)

How It Works

Gravitational potential energy (E = mgh) is the energy stored in an object because of its height above a reference point. The higher or heavier the object, the more potential energy it has. When released, this energy converts to kinetic energy as the object falls. Standard gravity on Earth is 9.80665 m/s².

Example Problem

A 10 kg rock sits on a cliff 30 m above the ground. How much potential energy does it have?

  1. Identify the known values: mass m = 10 kg, gravitational acceleration g = 9.81 m/s², height h = 30 m.
  2. Determine what we are solving for: the gravitational potential energy E stored in the rock relative to the ground.
  3. Write the potential energy equation: E = m × g × h.
  4. Substitute the known values: E = 10 kg × 9.81 m/s² × 30 m.
  5. Compute the result: E = 2,943 J (joules).
  6. Interpret the answer: if the rock falls, all 2,943 J converts to kinetic energy just before impact (ignoring air resistance). This is equivalent to the work gravity does on the rock over 30 m.

A simpler example: a 5 kg object at 10 m height has E = 5 × 9.81 × 10 = 490.5 J of potential energy.

When to Use Each Variable

  • Solve for Energy — when you know mass, gravity, and height — e.g., calculating the stored energy in a water reservoir or raised weight.
  • Solve for Mass — when you know the energy, gravity, and height — e.g., determining what mass was lifted to a known height using a known amount of energy.
  • Solve for Gravity — when you know energy, mass, and height — e.g., back-calculating gravitational acceleration on another planet.
  • Solve for Height — when you know energy, mass, and gravity — e.g., finding how high an object was raised given the work done.

Key Concepts

Potential energy depends on the choice of reference point (where h = 0). Common choices are the ground, the floor, or the lowest point in the problem. Only differences in potential energy matter physically. In a closed system, the total mechanical energy (KE + PE) is conserved.

Applications

  • Hydroelectric power: calculating the energy available from water stored at elevation in a reservoir
  • Mechanical engineering: sizing counterweights in elevators and crane systems
  • Roller coaster design: determining maximum speed at the bottom of a hill from the initial height
  • Space science: computing the energy needed to lift a satellite to its orbital altitude

Common Mistakes

  • Forgetting to convert height to consistent units — mixing feet and meters or using centimeters instead of meters will give answers off by orders of magnitude
  • Using the wrong reference point and then comparing energies from different references — only differences in PE are physically meaningful
  • Ignoring that g varies with altitude and location — using 9.81 m/s² is adequate for Earth's surface but not for high-altitude or planetary calculations

Frequently Asked Questions

How do I find the stored energy of an object at a height?

Use the formula E = m × g × h. Multiply the object's mass (in kilograms) by the acceleration due to gravity (9.81 m/s² on Earth) by the height above your reference point (in meters). The result is energy in joules. For example, a 10 kg object at 5 m height has E = 10 × 9.81 × 5 = 490.5 J.

What variables determine gravitational potential energy?

The gravitational potential energy formula is E = mgh, where E is energy in joules, m is mass in kilograms, g is gravitational acceleration in m/s², and h is height in meters. You can rearrange it to find mass (m = E/gh), gravity (g = E/mh), or height (h = E/mg).

What is the difference between potential energy and kinetic energy?

Potential energy is stored energy due to an object's position (height), while kinetic energy is the energy of motion (½mv²). As an object falls, its potential energy converts to kinetic energy. At the highest point PE is maximum and KE is zero; at the lowest point KE is maximum and PE is zero (assuming no friction).

What is the zero reference point for potential energy?

You choose the reference point (where h = 0) based on convenience — commonly the ground, the floor, or the lowest point in the problem. Only differences in potential energy are physically meaningful, so the choice of reference does not affect the physics. Changing the reference shifts all PE values by the same constant.

Does potential energy depend on the path taken?

No. Gravitational potential energy depends only on the object's height, not on the path it took to get there. Whether you carry a box straight up a ladder or up a winding ramp, the change in potential energy is the same. Gravity is a conservative force, which means the work it does depends only on the starting and ending positions.

Can potential energy be negative?

Yes, if the object is below your chosen reference point. For example, a ball in a valley with the hilltop as the reference has negative potential energy. The sign depends entirely on where you set h = 0. Negative PE simply means the object is below the reference — it has no special physical significance beyond that.

How does gravity affect potential energy on other planets?

Potential energy scales directly with gravitational acceleration (g). On the Moon (g ≈ 1.62 m/s²), the same mass at the same height has about one-sixth the potential energy it would on Earth (g ≈ 9.81 m/s²). On Jupiter (g ≈ 24.8 m/s²), it would have roughly 2.5 times more. Use this calculator and change the gravity value to compare.

Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Potential Energy Formula

Gravitational potential energy measures the energy stored in an object due to its position above a reference point:

E = m × g × h

Where:

  • E — potential energy, measured in joules (J)
  • m — mass of the object, measured in kilograms (kg)
  • g — acceleration due to gravity, measured in meters per second squared (m/s²). Standard Earth gravity is 9.80665 m/s².
  • h — height above the reference point, measured in meters (m)

The formula assumes a uniform gravitational field (constant g) and measures height relative to a chosen reference point. It applies to any object near a planet's surface where g does not vary significantly with altitude.

Worked Examples

Civil Engineering

A water tower holds 50,000 kg of water at 30 m elevation. What is the gravitational potential energy?

Water towers store potential energy that converts to pressure head for distribution. Calculate the energy stored in the elevated water mass.

  • Mass: m = 50,000 kg
  • Gravity: g = 9.81 m/s²
  • Height: h = 30 m
  • E = 50,000 × 9.81 × 30
  • E = 14,715,000 J ≈ 14.7 MJ

This stored energy provides the pressure needed to push water through distribution pipes without pumps during normal demand periods.

Hydropower

A dam has a 100 m head with 200 kg/s of water flow. What is the potential energy per second?

Hydroelectric plants convert the potential energy of water at elevation into electricity. For a single second of flow at 200 kg:

  • Mass: m = 200 kg (one second of flow)
  • Gravity: g = 9.81 m/s²
  • Height: h = 100 m
  • E = 200 × 9.81 × 100
  • E = 196,200 J/s ≈ 196 kW

Real turbine efficiency is 85-95%, so actual electrical output would be roughly 167-186 kW from this flow rate and head.

Rock Climbing

A 75 kg climber is 15 m above the last protection point. What potential energy would be released in a fall?

When a lead climber falls, gravitational potential energy converts to kinetic energy absorbed by the rope and protection system.

  • Mass: m = 75 kg
  • Gravity: g = 9.81 m/s²
  • Height: h = 15 m
  • E = 75 × 9.81 × 15
  • E = 11,036.25 J ≈ 11 kJ

Modern dynamic climbing ropes are designed to absorb this energy over a controlled distance, reducing the peak force on the climber to safe levels (under 12 kN per UIAA standards).

Related Calculators

Related Sites

Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.