Potential energy calculator

How do I calculate gravitational potential energy?

PE = m × g × h. Lifting 10 kg by 5 m stores 10 × 9.80665 × 5 = 490.3 joules.

What value of g does it use?

Standard gravity, 9.80665 m/s². You can change it — the Moon's is about a sixth of Earth's.

Where is height measured from?

From any level you choose as zero — the floor, the ground, a table. Only differences in potential energy matter.

How is it related to kinetic energy?

Without friction, falling turns potential energy into kinetic energy: 490.3 J of PE becomes 490.3 J of KE at the bottom.

How do I convert joules to foot-pounds?

Divide by 1.3558: 490.3 J is 361.6 ft·lbf.

Can potential energy be negative?

Yes, if the object is below the level you chose as zero. OpenStax University Physics (Volume 1, Example 8.2) puts the zero at the base of a mountain: a 75 kg hiker at the summit, 147 m higher, has 108 kJ, and the same hiker at sea level, 48 m below the base, has −35.3 kJ. The sign depends on the zero you pick; only differences in potential energy have physical meaning.

How fast is an object moving after it falls a given height?

Without friction or air resistance, the lost potential energy becomes kinetic energy: mgh = ½mv², so v = √(2gh) and the mass cancels. After a 3 ft (0.9144 m) drop that is 4.23 m/s, or 9.5 mph, for any object. The free fall calculator also gives the time.

What about the energy stored in a spring?

A stretched or compressed spring stores elastic potential energy U = ½kx², where k is the spring constant and x the displacement from the relaxed length (OpenStax University Physics, Volume 1, section 8.1, Eq. 8.7). The Hooke’s law calculator gives it in joules.

The formula and what it assumes

Near the surface of the Earth the gravitational potential energy of a mass m at height y is U = mgy (OpenStax University Physics, Volume 1, section 8.1, Eq. 8.5), and the change when it moves from one height to another is mg times the change in height (Eq. 8.4). The formula assumes g is constant, which is accurate for the heights of ordinary life but not for hundreds of kilometers: at 400 km altitude g is about 88% of its surface value (section 13.2). The zero of height is your choice, but you must state it and keep it for the whole problem.

Worked example: lifting a 40 lb bag by 3 ft

The mass is 40 × 0.45359237 = 18.14 kg and the rise is 3 × 0.3048 = 0.9144 m. Then U = 18.14 × 9.80665 × 0.9144 = 162.7 J, which is 120.0 ft·lbf. That agrees with the quick rule in US units: a weight of 40 lbf lifted 3 ft is 40 × 3 = 120 ft·lbf. On the Moon, with g = 1.62 m/s², the same lift stores 26.9 J.

Common mistakes

Using the length of a slope instead of the vertical rise is the classic one: a 10 m ramp that rises 2 m changes the energy by mg × 2 m, not mg × 10 m. Another is counting the weight twice: from a weight in newtons the energy is weight × height, from a mass it is mass × g × height. In the calculator, “lb” is a mass, so 40 lb needs no further conversion. And do not mix the unit of g with the unit of height: feet with 9.80665 m/s² is off by a factor of 3.28.

Sources: OpenStax University Physics, Volume 1, section 8.1 (potential energy of a system, Eq. 8.4, 8.5 and 8.7; Example 8.2) and section 13.2 (gravitation near the Earth’s surface; Example 13.4); standard gravity 9.80665 m/s², the pound (0.45359237 kg) and the foot-pound force (1.355818 J) are from NIST SP 811, Appendix B.8; the Moon’s 1.62 m/s² is from the NASA Planetary Fact Sheet. Related: kinetic energy calculator, free fall calculator, Hooke’s law calculator and work calculator.