A construction worker drops a 2 kg wrench from a 15 meter scaffold. How much energy is stored in that wrench the instant before it hits the ground? That question is a potential energy problem. Potential energy is stored energy due to an object's position or configuration. The two most common forms are gravitational potential energy, given by PE = mgh, and elastic potential energy, stored in a stretched or compressed spring, given by PE = 1/2 kx^2. The SI unit of energy is the joule (J), where 1 J = 1 kg x m^2/s^2 = 1 N x m. The concept of energy conservation has a rich history. Emilie du Chatelet, an 18th century French physicist and mathematician, translated Newton's Principia and argued that energy is proportional to velocity squared, not velocity. Her work helped establish the principle that energy is conserved, only transformed between forms. This calculator handles both gravitational and elastic potential energy, and it solves for any variable in each formula.
What This Calculator Does
The calculator has two modes. Gravitational mode uses PE = mgh and solves for energy, mass, height, or gravity. Elastic mode uses PE = 1/2 kx^2 and solves for energy, spring constant, or displacement. You pick the mode, choose which variable to solve for, then enter the known values with their units. The calculator converts every input to SI units, runs the formula, and shows the result. When you solve for energy, it also displays conversions to calories and watt-hours. Supported mass units are kg, g, lb, and metric tons. Supported height and displacement units are m, cm, ft, and in. Supported energy units include joules, kilojoules, calories, kilocalories, watt-hours, kilowatt-hours, foot-pounds, and BTUs. Spring constant units include N/m, N/cm, lbf/in, and lbf/ft.
For related physics calculations, try our Kinetic Energy Calculator to find the energy of a moving object, our Force Calculator to compute force from mass and acceleration, or our Work Calculator to calculate mechanical work from force and distance.
Inputs Required
- Mass: The object's mass (gravitational mode, not needed when solving for mass)
- Height: The vertical position above the reference point (gravitational mode, not needed when solving for height)
- Gravity: The gravitational acceleration, defaulting to 9.80665 m/s^2 (gravitational mode, not needed when solving for gravity)
- Spring Constant: The stiffness of the spring (elastic mode, not needed when solving for spring constant)
- Displacement: The distance the spring is stretched or compressed from rest (elastic mode, not needed when solving for displacement)
- Potential Energy: The energy value (not needed when solving for energy)
Outputs Provided
- Primary result: The solved variable in SI units
- Energy conversions: Joules, calories, and watt-hours (when solving for energy)
- Calculation breakdown: Shows the formula with your substituted values
How the Calculation Works
PE = m x g x h
m = PE / (g x h)
h = PE / (m x g)
g = PE / (m x h)
PE = 1/2 x k x x^2
k = 2 x PE / x^2
x = sqrt(2 x PE / k)
1 J = 1 kg x m^2/s^2
1 cal = 4.184 J
1 kWh = 3,600,000 J
1 BTU = 1055.06 J
The calculator converts mass to kilograms, height and displacement to meters, spring constant to newtons per meter, and energy to joules. Then it applies the right rearrangement of the formula. For gravitational energy, it multiplies mass, gravity, and height. For elastic energy, it takes half the spring constant times the displacement squared. When solving for energy, the result is also converted to calories (divide by 4.184) and watt-hours (divide by 3600) for practical context. One thing to notice: gravitational potential energy scales linearly with all three variables. Double the height and you double the energy. Elastic potential energy scales with the square of displacement, so doubling the stretch quadruples the stored energy.
How to Use the Calculator
- Pick a mode: Gravitational PE or Elastic PE
- Choose which variable to solve for from the dropdown
- Enter the known values with their units
- Read the primary result in the highlighted box
- Review the calculation breakdown and any energy conversions
- Copy the result if you need it elsewhere
Example Calculations
Maria lifts a 12 kg box onto a shelf 2.5 meters above the floor. How much gravitational potential energy does the box gain?
- PE = 12 kg x 9.80665 m/s^2 x 2.5 m = 294.20 J
- That equals about 70.3 calories or 0.0000817 kWh
In a second example, David compresses a spring with a constant of 800 N/m by 0.05 meters. What is the elastic potential energy?
- PE = 0.5 x 800 x (0.05)^2 = 0.5 x 800 x 0.0025 = 1.0 J
- That equals about 0.239 calories or 0.000278 Wh
Real-World Scenarios
Hydroelectric Dam Engineer at Hoover Dam
Sarah is a hydroelectric engineer working at Hoover Dam on the Arizona-Nevada border. She needs to estimate the gravitational potential energy of water in the reservoir. The water at the surface sits about 180 meters above the turbines. A cubic meter of water has a mass of about 1,000 kg. For a single cubic meter, PE = 1,000 kg x 9.80665 m/s^2 x 180 m = 1,765,197 J, or about 1.77 MJ. The dam's generators process roughly 300 cubic meters per second at full flow. That means the available potential energy per second is 1,765,197 x 300 = 529,559,100 J/s, or about 530 MW of power before efficiency losses. Real generators convert about 90% of this to electricity, giving roughly 477 MW. Sarah uses these numbers to plan turbine maintenance schedules and match output to grid demand.
Bungee Jumping Design in Queenstown, New Zealand
James designs bungee jumping cords for a company in Queenstown, New Zealand. A jumper named Tom weighs 75 kg and jumps from a platform 43 meters above the Kawarau River. James treats the bungee cord as a spring with an effective spring constant of 120 N/m. He wants to find the maximum stretch so the cord stops Tom before he hits the water. At the top, Tom has gravitational PE = 75 x 9.80665 x 43 = 31,626 J. At the bottom of the jump, all of that energy becomes elastic PE in the cord. Using PE = 1/2 k x^2, James solves for displacement: x = sqrt(2 x 31,626 / 120) = sqrt(527.1) = 22.96 m. So the cord stretches about 23 meters. James adds a safety margin and chooses a cord length plus stretch that keeps Tom at least 5 meters above the water.
Elevator Counterweight System in Chicago
Linda is a mechanical engineer designing an elevator system for a 40-story office building in Chicago. The elevator car has a mass of 1,200 kg. The counterweight has a mass of 1,600 kg, which balances the car plus a typical passenger load. Linda calculates the gravitational potential energy difference as the car travels from the ground floor to the 40th floor, about 150 meters up. The counterweight moves down 150 meters at the same time. The car gains PE = 1,200 x 9.80665 x 150 = 1,765,197 J. The counterweight loses PE = 1,600 x 9.80665 x 150 = 2,353,596 J. The net change is 2,353,596 - 1,765,197 = 588,399 J. This net energy is what the motor must handle, and because the counterweight is heavier, the motor actually does negative work (it brakes) when the car goes up empty. Linda uses these figures to size the motor and the braking resistor.
Common Mistakes to Avoid
- Mixing up the reference point for height: Gravitational PE depends on the height you choose as zero. Only differences in PE matter physically. Always state your reference level before reporting a value, and keep it consistent across a problem.
- Forgetting that displacement is squared in elastic PE: The elastic formula uses x^2, not x. Doubling the stretch quadruples the stored energy. People often treat it as linear and get the wrong answer by a factor of two or four.
- Using the wrong gravity value: Standard Earth gravity is 9.80665 m/s^2. The value 9.8 or 10 is fine for rough estimates, but for precise work use the full value. On the Moon gravity is 1.62 m/s^2 and on Mars it is 3.71 m/s^2. Using 9.8 on the Moon gives a result about six times too large.
- Not converting units before calculating: If you enter mass in pounds and height in feet without converting, you will not get joules. This calculator handles conversions automatically. For hand calculations, convert everything to SI units first.
Limitations of This Calculator
This calculator assumes a uniform gravitational field, where g is constant. In reality, gravity decreases with altitude. At 400 km up, where the International Space Station orbits, gravity is about 8.67 m/s^2, not 9.81. For most everyday heights this difference is negligible, but for orbital mechanics or tall structures you need the inverse square law. The elastic mode assumes a linear spring that follows Hooke's law, F = kx. Real springs have a linear region, but beyond it they deform permanently or behave non-linearly. This calculator does not account for spring fatigue, damping, or non-linear stiffness. It also ignores relativistic effects. At speeds or energy scales approaching the speed of light, classical formulas break down and you need relativistic energy equations. For everyday engineering, physics homework, and most practical problems, the classical formulas here are accurate to many decimal places.
Authoritative Research and Resources
- NIST: SI Units Reference - The National Institute of Standards and Technology defines the joule and all SI units. This page gives the official definition of the joule as one newton-meter, which grounds every calculation on this page in a recognized standard.
- HyperPhysics: Potential Energy - Hosted by Georgia State University, HyperPhysics offers concise explanations of gravitational and elastic potential energy with diagrams. It is useful for seeing how PE relates to work and kinetic energy through conservation of energy.
- Khan Academy: Work and Energy - A free resource with video lessons and practice problems covering potential energy, kinetic energy, the work-energy theorem, and conservation of energy. It is a good starting point if you are learning these concepts for the first time.