Potential Energy Calculator

PE = mgh. Gravitational potential energy is mass times the acceleration of gravity times height. Kilograms, meters per second squared and meters give joules.

Worked example, the one above: a 70 kg person climbing 10 m. PE = 70 × 9.80665 × 10 = 6,865 J, 6.865 kJ. In food terms that is 1.64 kcal, a strikingly small number. Climbing stairs burns more than this because the body is far from perfectly efficient, so it spends several times the mechanical work as heat to deliver it.

Height from what? Potential energy is measured from a reference level you choose: the floor, the ground, the bottom of the hill. Only changes in it mean anything physically, so the choice is free as long as you keep it. Below the reference, h is negative and so is the potential energy. The calculator accepts a negative height for exactly that reason.

Why it is m × g × h. Lifting an object at a steady speed takes a force equal to its weight, mg, applied through the height h. Work is force times distance, so the work done is mgh, and it is stored: let the object fall and the same energy comes back as kinetic energy. Setting mgh = ½mv² gives the speed after falling h, v = √(2gh), and the mass cancels. That is why every object falls at the same rate when air resistance is negligible. The calculator shows that speed for the height you enter.

On the Moon and Mars. The same lift on the Moon, where NASA gives surface gravity as 1.62 m/s², stores 1.134 kJ, about a sixth as much. On Mars, at 3.73 m/s², it is 2.611 kJ. The table under the result compares all three for your mass and height. Mass is the same everywhere; it is g that changes, and with it the weight and the energy of every lift.

When mgh stops working. The formula assumes g is constant over the whole height. Near a planet's surface that is excellent: g falls by only about 0.03% per kilometer of altitude on Earth. For satellites, rockets and anything whose height is a real fraction of the planet's radius, g changes appreciably along the way, and the correct form is the full gravitational potential energy, −GMm/r.

Potential energy is one kind of stored energy among several. A compressed spring holds elastic potential energy, ½kx²; a charged capacitor and a battery hold electrical energy. This calculator covers the gravitational kind only.