Lift something and you do work against gravity; that work is stored as gravitational potential energy and comes back as motion when the object falls. This calculator puts a number on it with U = mgh, and it also runs the formula backward: give it an energy and it returns the mass, height or gravitational acceleration that would produce it. It suits homework problems, rough checks on hoists and drop tests, and “what if this happened on the Moon” questions alike.
How to use the potential energy calculator
- Under Solve for, choose Energy, Mass, Gravity or Height. The box for that quantity disappears and a Show the result in menu lets you pick its unit.
- Enter the Mass (m) in kg, g, mg, metric tons, lb, oz, US tons or stone.
- Keep Gravitational acceleration (g) at 9.80665 m/s² for Earth, or type another value in m/s², ft/s² or multiples of g.
- Enter the Height (h) above the level you are calling zero. Use a negative number for a position below it.
- Read the result with its conversions (kJ, cal, ft·lbf and Wh for energy), the object’s Weight (mg) in newtons and, for positive heights, the Speed if dropped in m/s, km/h and mph. “Show the work” lists each substitution.
Potential energy formula
With mass in kilograms, g in m/s² and height in meters, the product comes out in joules. The calculator rearranges the same equation for the other three unknowns:
Mass and g must be positive. Height and energy may be negative, but they must share a sign when you solve for mass or g, and a height of zero is rejected there because it would mean dividing by zero. The drop speed comes from equating the stored energy with kinetic energy:
Worked example
A 2 kg toolbag hauled up 10 m to a roof
U = 2 kg × 9.80665 m/s² × 10 m = 196.1 J, which is about 0.196 kJ, 46.9 cal or 144.7 ft·lbf.
The bag weighs 2 × 9.80665 = 19.61 N. If it slid off the edge, it would hit the ground at √(2 × 9.80665 × 10) ≈ 14.0 m/s, about 50 km/h, ignoring drag.
Change g to 1.62 m/s² and the same bag on the Moon stores only 32.4 J and lands at 5.69 m/s. US units work just as directly: because one pound-force is defined as the weight of one pound under standard gravity, a 10 lb weight raised 6 ft stores exactly 60 ft·lbf.
Choosing the zero level
There is no absolute zero of gravitational potential energy near the ground. You can measure height from the floor, the tabletop, sea level or the bottom of a mine shaft, and U will change with that choice. What never changes is the difference between two positions, and that difference is the only thing that turns into motion, heat or electricity. A book dropped from a 2 m shelf to the floor releases mg × 2 m whether you call the floor zero or set zero at the street 3 m below.
Height is taken as positive upward. Below your chosen zero, h and U are negative, which simply means the object would need energy supplied to reach the reference level. The calculator’s note under the result is a reminder of exactly this.
Where U = mgh breaks down
The formula assumes g stays constant over the height range, and near the ground it nearly does. Further out, gravity weakens with the square of the distance from Earth’s center:
| Altitude | Local g | Share of surface value |
|---|---|---|
| 10 km (airliner) | 9.78 m/s² | 99.7% |
| 100 km (edge of space) | 9.51 m/s² | 96.9% |
| 400 km (space station) | 8.68 m/s² | 88.5% |
Above about 100 km the calculator adds a note suggesting U = −GMm/r, which sets zero at infinite distance and is always negative for a bound object. A telling case is solving for height with 1 kWh and 1 kg: mgh asks for about 367 km, well inside the range where the simple formula is already off by several percent. It also shows how little energy gravity stores per kilogram. Lifting a full tonne through 100 m stores 980.7 kJ, or only about 0.27 kWh, which is why pumped-storage hydro plants move enormous volumes of water.
Related calculators
To follow the energy once the object is moving, use the kinetic energy calculator or the free fall calculator. The energy stored in a stretched spring has its own formula on the elastic potential energy calculator.
Frequently asked questions
What is the formula for gravitational potential energy?
U = mgh: mass in kilograms times gravitational acceleration in m/s² times height in meters gives the energy in joules. Lifting 2 kg by 10 m on Earth stores 2 × 9.80665 × 10 ≈ 196.1 J.
Can potential energy be negative?
Yes. U is measured from whatever level you decide to call zero, so an object below that level has negative potential energy. The calculator accepts negative heights; a 2 kg mass 5 m below the reference has U ≈ −98.07 J.
Does the speed of a falling object depend on its mass?
Not when air resistance is negligible. Setting mgh equal to ½mv² cancels the mass and leaves v = √(2gh), so anything dropped from 10 m reaches about 14.0 m/s (roughly 50 km/h or 31 mph), whether it is a pebble or a piano.
What value of g does the calculator use?
It starts at standard gravity, 9.80665 m/s², the conventional value fixed by the CGPM in 1901. Real surface gravity ranges from about 9.78 m/s² at the equator to 9.83 m/s² at the poles, and you can enter any value, such as 1.62 m/s² for the Moon or about 3.71 m/s² for Mars.
When does U = mgh stop being accurate?
When the height becomes a noticeable fraction of Earth's radius. Gravity is about 3% weaker 100 km up, and at 400 km the simple formula overstates the energy needed to lift each kilogram by roughly 6%. For orbital heights use U = −GMm/r instead.