Gravitational potential energy

03How it works

Why this calculation

Lift a brick onto a shelf and you have done work against gravity; that work is now stored in the brick as gravitational potential energy (PE). Drop the brick and the energy converts to kinetic energy on the way down — quickly enough that a brick from second-storey height is dangerous. Engineers, physicists, hydropower planners, climbers, roller-coaster designers, and crane operators all rely on the same simple expression: PE = m · g · h. Knowing it lets you compare a falling object's destructive impact to other energy units (kilocalories of food, kilowatt-hours of electricity), size pumped-storage hydro reservoirs, predict a climber's free-fall rope load on a fall, and budget the kinetic energy a roller coaster wagon will deliver at the bottom of a drop. The formula scales from millijoules (a falling penny) to terajoules (a hydro dam reservoir), and the only knob to turn besides mass and height is the local gravitational acceleration g.

The formula

PE = m · g · h

m is the mass in kilograms (or pounds — the calc converts).
g is the local gravitational acceleration, ≈ 9.81 m/s² on Earth at sea level. Different on every other body in the solar system.
h is the height above some reference (often the ground, but you choose).

The result is in joules (J), the SI unit of energy. Conversions: 1 kJ = 1000 J; 1 kcal = 4184 J; 1 kWh = 3 600 000 J; 1 ft·lb = 1.356 J. From the same starting condition you can predict free-fall impact velocity (no air drag): v_impact = √(2·g·h).

How to use it

Pick the unit system (SI = kg, metres, m/s²; Imperial = lb, feet, ft/s²). Enter the mass and the height drop (or rise — the formula is symmetric in sign). Pick a planet to set g — Earth, Moon, Mars, the Sun, or any of the major bodies. If your scenario is exotic (lunar lander, science-fiction asteroid, custom centrifuge), pick Custom and type your own g.

The big number is energy in joules. The kJ, kcal, kWh, and ft·lb readouts let you compare to familiar quantities: 50 kg × 10 m × 9.81 = 4905 J ≈ 1.17 kcal — about the energy of one peanut. The free-fall impact speed assumes vacuum; in air, real impact is slower because of drag.

Worked example

A 100 kg rock falls off a 10 m cliff on Earth: PE = 100 × 9.81 × 10 = 9 810 J ≈ 9.8 kJ ≈ 2.34 kcal ≈ 0.0027 kWh. Free-fall speed at impact: v = √(2 × 9.81 × 10) = 14.0 m/s ≈ 50 km/h — definitely fatal if it landed on you. The same rock dropped from the same height on the Moon (g = 1.62 m/s²) stores only 1620 J and lands at 5.7 m/s — survivable. Same rock from a 50 m building on Earth: PE = 49 050 J, impact 31.3 m/s ≈ 113 km/h, the stuff of action movies.

Pitfalls

PE is relative to a chosen zero. The calc lets you set the reference; what matters physically is change in height, not absolute height.
g varies by location even on Earth — 9.78 at the equator vs 9.83 at the poles, 0.3% lower at altitude per 10 km. For high-precision work in geophysics or pendulum clocks, this matters.
Mass vs weight — kg is mass; weight (the force gravity pulls on it) is m·g in newtons. Confusing them gives wrong answers.
Air drag — for objects falling more than a few meters, especially light ones (feathers, plastic bags), drag converts much of the PE to heat in the surrounding air rather than to kinetic energy at impact.
Non-uniform field — at altitudes much greater than the planet radius, g decreases significantly; use the inverse-square form (-G·M·m/r) instead.
PE depends on path, not on details in a conservative field — but the calc only handles vertical height; for inclined paths, only the vertical component matters.
Stored vs released — PE is potential. The energy is only available if something releases it (a brake fails, a spring snaps, a rope cuts). Don't conflate energy with hazard.

Variations

This calc covers gravitational PE in a uniform field — by far the most common case. Other forms of PE include: elastic PE ½kx² in springs (compression/stretch from rest), electrical PE qV in capacitors and battery cells (charge × voltage), chemical PE stored in molecular bonds (gasoline ~46 MJ/kg, fat ~37 MJ/kg, dynamite ~5 MJ/kg), nuclear PE in atomic nuclei (uranium fission ~80 TJ/kg). The inverse-square gravitational form -G·M·m/r is needed for orbital mechanics and escape-velocity calculations. In rotating reference frames, centrifugal PE adds ½ω²r² (think of a centrifuge rotor or the Coriolis-corrected horizontal pendulum). For pendulum mechanics, PE at the apex (m·g·h above the lowest point) converts entirely to KE at the bottom (½·m·v²) — the basis for the simple pendulum equation T = 2π√(L/g).