Gravitational Potential Energy Calculator

Height is work you can get back

PE = mgh is not a separate law. It is the work done against gravity, written in a form you can look up. Raising a mass steadily needs an upward force equal to its weight, mg, and carrying that force through a height h costs mgh of work. Gravity is a conservative force, so none of that work leaks away: it sits in the arrangement of the mass and the Earth until the mass is allowed to fall. This is also why the formula has no term for the route taken. A winding ramp and a vertical shaft to the same landing store exactly the same energy.

The g in the middle is the local field strength, taken as 9.80665 m/s² by convention on Earth. Swap it and the same lift costs something quite different: 1.625 on the Moon, 3.721 on Mars and 24.79 m/s² at the cloud tops of Jupiter. Shelving a crate on Mars therefore takes about 38 per cent of the energy it takes here, and on Jupiter about 2.53 times as much.

Worked example

A 2 kg mass is raised 10 m on Earth.

• PE = m g h = 2 × 9.80665 × 10
• PE = 196.133 J
• Released, all of it becomes kinetic: mgh = 0.5 m v²
• v = √(2gh) = √(2 × 9.80665 × 10) = √196.133
• v = 14.00 m/s

That impact speed is 50.42 km/h, and the stored energy is 46.88 cal. Food labels count in kilocalories, so on a label that is 0.047 kcal. The two numbers 196.133 J and 196.133 m²/s² look suspiciously alike, and the reason is arithmetic rather than physics: with a mass of exactly 2 kg the factor 0.5 m equals 1, so the joule figure and the square of the speed happen to coincide. Change the mass to 5 kg and the energy rises to 490.3 J while the impact speed stays at 14.00 m/s.

The fall does not care about the mass

Set mgh = 0.5 m v² and the mass cancels from both sides, leaving v = √(2gh). Drop a 2 kg ball and a 200 kg ball from 10 m in a vacuum and both arrive at 14.00 m/s. Real air spoils this, and it spoils it in favour of the heavier object, because drag depends on frontal area and speed rather than mass. For a human body terminal speed is around 55 m/s, so past about 150 m of drop the square root formula starts to overstate the arrival speed badly.

The same equation runs the largest energy stores on the grid. Pumped storage hydroelectricity lifts water into a high reservoir when power is cheap and lets it back down through turbines when it is not. One tonne of water raised 300 m holds 1000 × 9.80665 × 300 = 2.942 MJ, which is 0.817 kWh, enough to run a 100 W lamp for eight hours. The volumes make the scheme work: a reservoir holding a million tonnes at that head stores over 800 MWh.

Common mistakes

• Measuring the height along the slope. Only the vertical rise belongs in h. A 5 m ramp inclined at 30 degrees lifts a load 2.5 m, so it stores half the energy a careless reading suggests.
• Leaving out g altogether. Mass times height has units of kilogram metres, which is not energy. The 9.80665 is what converts a mass into the weight you are actually lifting.
• Treating the zero of height as fixed. A book on a desk has one potential energy measured from the desk and another measured from the floor, and both are correct. Only the change carries physical meaning, so choose a reference, state it, and keep it for the whole problem.
• Trusting the impact speed where air matters. The √(2gh) readout assumes free fall. It is close for a dense object over a few metres and useless for a leaf, a parachute or any long drop.

Converting units first? Use the mass, energy, length and acceleration conversion tables.