Potential Energy Calculator
Gravitational potential energy near a surface or between two distances from a planet, with unit conversions and a chart.
🏔️ What is Potential Energy?
Potential energy is energy stored in an object because of where it is. Lift a book onto a shelf and you do work against gravity; that work does not vanish, it is stored as gravitational potential energy and can be released later when the book falls. Near the surface of a planet the stored energy is U = m g h: the mass of the object, the gravitational acceleration, and the height above a reference level that you choose. The SI unit is the joule.
The idea matters everywhere energy is moved against gravity. A diver on a platform, water behind a dam, a load raised by a crane, a skier at the top of a lift and a rocket leaving the ground all gain or lose potential energy. Pumped hydro plants store electricity by pushing water uphill, and the amount they can store is exactly m g h. Engineers use it to size cranes and lifts, physics students use it with kinetic energy to solve conservation problems, and anyone curious about spaceflight uses the orbital form to see how much energy a launch must supply.
Two points cause most confusion. First, potential energy is relative. The value of m g h depends on where you put h = 0, so a book on a shelf has potential energy relative to the floor but a different value relative to the basement. Only differences are physical, which is why this page also reports the change in U between two heights. Second, m g h is an approximation. It assumes gravity is the same at every height, which is excellent for a few hundred metres but not for a climb into orbit. For distances comparable to the planet radius the exact expression is U = -G M m / r, with zero at infinity, and it is always negative for a bound object. See the kinetic energy calculator for the energy of motion that potential energy turns into, and the work, energy and power calculator for a quick one-line version of m g h alongside work and power.
This calculator is the dedicated deep version. It converts the result into joules, kilojoules, foot-pounds, kilowatt hours and calories, gives the speed the object would reach if all the energy became kinetic, finds the mass, height or energy from the other two, and compares the exact orbital formula with the m g h shortcut so you can see the error for yourself.
📐 Potential Energy Formulas
The shortcut and the exact formula agree for small climbs. Writing r2 = r1 + h gives ΔU = G M m h / (r1 r2), while m g h with g = G M / r1² gives G M m h / r1². Their ratio is r2 / r1, so the shortcut overestimates the energy by exactly h / r1, which is the figure shown as the shortcut error. See Gravitational energy on Wikipedia, HyperPhysics and OpenStax College Physics, section 7.3. For the energy needed to leave a planet entirely, use the escape velocity calculator.
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A 70 kg diver on a 10 m platform
Reference level is the water surface, g = 9.80665 m/s²
Example 2 — A 2 kg book on a 1.5 m shelf
Reference level is the floor
Example 3 — Pumped-storage water, 1000 kg raised 100 m
One tonne of water lifted to an upper reservoir
Example 4 — The same diver on the Moon
70 kg at 10 m with g = 1.62 m/s²
Example 5 — Height of a 25 kg load holding 5,000 J
Solve mode, find height on Earth
Example 6 — Lifting a 1000 kg satellite to 400 km: exact vs m g h
Earth M = 5.972 × 1024 kg, r1 = 6371 km (surface), r2 = 6771 km
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for gravitational potential energy?
Near a planet surface it is U = m g h, mass times gravitational acceleration times height above a chosen reference level. For a 70 kg person 10 m above the ground on Earth, U = 70 x 9.80665 x 10 = 6,864.65 J. For large distances use U = -G M m / r, with zero at infinity.
What are the units of potential energy?
The SI unit is the joule (J), where 1 J = 1 kg m squared per second squared. Other common units are the kilojoule, the kilowatt hour (1 kWh = 3.6 million J), the foot-pound (1 ft-lbf = 1.3558 J) and the thermochemical calorie (4.184 J).
Can potential energy be negative?
Yes. Potential energy is measured from a reference level you choose. An object below that level has a negative mgh. In the orbital formula U = -GMm/r the zero is at infinity, so every object in a bound orbit has negative potential energy.
How do I find height from potential energy?
Rearrange U = m g h to h = U / (m g). A 25 kg load holding 5,000 J on Earth sits 5,000 / (25 x 9.80665) = 20.39 m above the reference level. Choose the Solve mode and Find height to do this automatically.
How do I find mass from potential energy?
Use m = U / (g h). Energy and height must have the same sign, and the height cannot be zero, because U is zero at the reference level whatever the mass is. The Solve mode handles the unit conversions for you.
How much energy is 1000 kg raised 100 m?
On Earth, U = 1000 x 9.80665 x 100 = 980,665 J, which is about 980.7 kJ or 0.2724 kWh. This is the idea behind pumped hydro storage: moving water uphill stores energy that a turbine recovers when it runs back down.
Why is the zero of potential energy arbitrary?
Only the change in potential energy appears in physical results, such as the work done or the speed gained in a fall. Shifting the reference level adds the same constant to every value and cancels in any difference. Pick the level that makes your problem simplest.
What is the difference between potential and kinetic energy?
Potential energy is stored energy due to position, such as height in a gravitational field. Kinetic energy is the energy of motion, 1/2 m v squared. In a fall without air resistance potential energy converts into kinetic energy, so the speed after falling h is v = sqrt(2 g h).
When does mgh stop being accurate?
The shortcut assumes g is constant. The exact change from distance r1 to r2 is G M m (1/r1 - 1/r2), and m g h with g taken at r1 is too large by (r2 - r1) / r1. For a 400 km climb from Earth surface that is about 6.3 percent.
Is potential energy a vector?
No. Potential energy is a scalar with no direction. The force it produces does have a direction: gravity pulls toward lower potential energy, and the force equals minus the gradient of U.