How much energy does it take to lift something, and how much comes back when it falls? Enter a mass and a height for the answer in joules, kJ, MJ, kWh and kcal, or work backwards to either input.
Potential Energy Calculator - Ep = mgh in J, kJ, kWh
How much energy does it take to lift something, and how much comes back when it falls? Enter a mass and a height for the answer in joules, kJ, MJ, kWh and kcal, or work backwards to either input.
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The energy in a raised object, from an apple to a reservoir
Between an apple set on a kitchen counter and the water held behind a small dam there are nine orders of magnitude, and one formula covers the whole span. Give the calculator a mass and a height and it returns the gravitational potential energy in joules, along with the same figure in kilojoules, megajoules, kilowatt hours and kilocalories. Give it an energy instead and it returns whichever of the two you left out. A 75 kg adult who climbs 10 m has stored 7,355 J, which is 7.355 kJ, 0.0020431 kWh or 1.7579 kcal.
The quantity has a single sentence behind it: the energy equals the mass times the gravitational acceleration times the height, and every consequence in this page follows from the fact that all three appear to the first power. Nothing is squared, nothing is inside a root, so doubling any one of them doubles the answer and doubling two of them quadruples it. That is a gentler relationship than the one governing a moving object, where speed is squared, and it is the reason a very heavy object barely off the ground routinely outranks a light one high in the air.
The tool also runs the formula backwards. Give it an energy and a height and it returns the mass that would hold that energy; give it an energy and a mass and it returns how high the object would have to go. Both are the same equation rearranged, and both are checked against the forward calculation before anything is published, so a figure taken out of one mode and put back into another closes the loop rather than drifting.
Three numbers, and which one you are missing
The form holds three quantities and asks for two of them. Pick the one you want first, because that choice decides which fields appear.
- What you want to work out - the energy, the mass or the height. Nothing else is shown until this is set, and no option is preselected, so the form cannot make the choice look as though you already made it.
- Mass - in kilograms, grams, metric tons or pounds. Pounds are converted at the international value of 0.45359237 kg, so an input in pounds and the same weight in kilograms give one identical answer rather than two that nearly agree.
- Height - in meters, centimeters, kilometers or feet, at 0.3048 m to the foot. This is the vertical distance between where the object is and wherever you decided to call zero, not the height above sea level.
- Energy - only when you are solving for mass or height. Joules, kilojoules, megajoules, kilowatt hours and kilocalories are all accepted, at 3,600,000 J to the kilowatt hour and 4,184 J to the kilocalorie, which is the calorie printed on food labels.
- Gravitational acceleration - optional. Left blank it uses standard gravity, 9.80665 m/s². Fill it in only if you are working somewhere that is genuinely not the Earth's surface, and the result will say how the answer compares.
- Read the answer. The headline gives the quantity you asked for, the line beneath restates the energy in five units, and the two tables place it against things that have been lifted and against everyday energies.
Height is the only thing that has to be agreed on
Potential energy is unusual among the quantities in physics in that its value is not fixed by the object alone. A mass on an upstairs windowsill has one potential energy relative to the floor of that room, a larger one relative to the ground outside, and a larger one still relative to the bottom of the basement. All three numbers are correct at the same time. What the formula actually measures is the work it took to move the object between two levels, and the second level is yours to choose.
This is why the field is labeled height rather than altitude. Nothing in the arithmetic knows or cares how far above sea level either point sits, only how far apart they are vertically. It is also why the result never changes when you carry an object horizontally: gravity pulls straight down, so a step sideways does no work against it, and a suitcase wheeled the length of an airport terminal arrives with exactly the energy it left with.
The other half of the formula is g, which the calculator treats as constant between the two levels. That is an approximation and it is a very good one near the ground. Gravity falls off with the square of the distance from the center of the Earth, so climbing a kilometer weakens it by about 0.03%, well below the precision of any mass you are likely to enter. The approximation only breaks down at the scale of orbits, where the energy has to be found from the full inverse-square expression instead.
Stored this way, the energy is genuinely recoverable. Let the object fall and every joule comes back as motion. That is the arithmetic behind pumped storage, behind a grandfather clock's weights, and behind the plain fact that a dropped tool hurts in proportion to how far it fell. From 10 m an object arrives at 14.00 m/s, which is 50.4 km/h or 31.3 mph, and the calculator prints that speed under the result.
What a lift costs, from a countertop to a dam
Every row below is a mass and a height written out, so any of them can be checked on a calculator: multiply the two and then by 9.80665. The same table appears under your result with your own figure slotted into the correct gap.
| Something lifted | Mass and height | Energy stored |
|---|---|---|
| An apple lifted onto a counter | 0.15 kg at 0.9 m | 1.32 J |
| A paperback put back on a shelf | 0.4 kg at 1.8 m | 7.06 J |
| A suitcase into an overhead bin | 20 kg at 1.2 m | 235.4 J |
| An adult up one flight of stairs | 75 kg at 3.3 m | 2.43 kJ |
| An adult up a ten-story building | 75 kg at 33 m | 24.27 kJ |
| A car raised onto an overpass | 1,400 kg at 8 m | 109.83 kJ |
| An adult to the Eiffel Tower top deck | 75 kg at 276 m | 203.00 kJ |
| An elevator with eight people, up 30 floors | 1,200 kg at 99 m | 1.17 MJ |
| A loaded truck over a mountain pass | 40,000 kg at 500 m | 196.13 MJ |
| A thousand metric tons of water behind a dam | 1,000,000 kg at 100 m | 980.66 MJ |
Two rows in that list are worth pausing on. A car raised only 8 m onto an overpass holds 109.83 kJ, while an adult who climbs four times higher, to the top of a ten-story building, holds 24.27 kJ. And the dam row is not a large dam. A thousand metric tons is a cube of water roughly 10 m on a side, and it still outweighs everything above it by three orders of magnitude.
The same joule, counted six ways
A figure in joules means very little on its own, so the second table under the result compares it against six energies most people have some feel for. Each one carries the arithmetic that produced it, which is the only way to make a comparison table checkable rather than decorative. The right-hand column shows what the 7,355 J from a 75 kg climb of 10 m is worth against each.
| Everyday energy | In joules | Where the figure comes from | A 10 m climb is worth |
|---|---|---|---|
| One gram of sugar, as food energy | 16,736 | 4 kcal at 4,184 J each | 0.4395 |
| A 10 W LED bulb for one hour | 36,000 | 10 W over 3,600 s | 0.2043 |
| A full smartphone battery | 62,370 | 4,500 mAh at 3.85 V, so 17.33 Wh | 0.1179 |
| Heating a mug of water for tea | 83,720 | 0.25 kg, 20 to 100 degrees, 4,186 J per kg per degree | 0.0879 |
| One slice of bread, as food energy | 334,720 | 80 kcal at 4,184 J each | 0.0220 |
| Boiling a full 1.7 L kettle | 569,296 | 1.7 kg, 20 to 100 degrees | 0.0129 |
Read down that column and the scale of gravitational storage becomes uncomfortable. Ten meters of climb, for a full-grown adult, is worth less than half a gram of sugar and about a fiftieth of a slice of bread. Run it the other way and one kilowatt hour, the unit a meter counts in, would lift 100 kg to 3,670.98 m, higher than any building ever constructed. Height is a very poor place to keep energy, and that is precisely why pumped storage is built at the scale of reservoirs rather than water tanks.
Why a heavy thing low down beats a light thing high up
Six things the formula quietly assumes
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Where the stored energy goes once the object is released, and how it grows with the square of the speed - See the calculator
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The weight you have to hold while lifting, which is the mass multiplied by the same gravitational acceleration - See the calculator
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What a falling object carries into whatever it lands on, and why that is not the same thing as its energy - See the calculator
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What the lift demands once a clock is attached to it, since the same energy over half the time needs twice the power - See the calculator