Potential energy is stored energy waiting to be released — the energy a boulder holds at the top of a hill or a drawn bow holds in its limbs. This calculator works out gravitational potential energy (PE = m·g·h) and elastic spring potential energy (PE = ½·k·x²), solving for whichever variable you leave blank, and links it to kinetic energy so you can turn a drop height into an impact speed. This guide explains both forms, why the choice of reference height matters, and how energy is conserved as things fall.
What potential energy is — and why it's relative
Potential energy is energy an object has because of where it is or how it is arranged, rather than how fast it is moving. Lift an object and you do work against gravity; that work is stored as gravitational PE and is released as the object falls. Compress a spring and the work you do is stored as elastic PE, released when the spring rebounds. A key subtlety with gravitational PE is that it is always measured relative to a chosen reference level: a book on a table has different PE depending on whether you measure height from the table, the floor, or the basement. Only differences in PE do physical work, so this calculator treats the height you enter as the height above your chosen zero.
Gravitational potential energy: PE = m·g·h
Near a planet's surface, gravitational potential energy is simply mass times gravity times height. The gravity term is what changes between worlds: g is 9.81 m/s² on Earth, 1.62 on the Moon, and 3.71 on Mars, so the same lift stores roughly six times less energy on the Moon than on Earth. This formula assumes g is constant, which is an excellent approximation near the surface but breaks down over distances comparable to a planet's radius — for satellites and escape-velocity problems you need the full gravitational potential −G·M·m/r instead. Within its range, PE = m·g·h is exact, and rearranging it lets you solve for the mass or height needed to store a target amount of energy.
Elastic potential energy and Hooke's law
An ideal spring obeys Hooke's law: the restoring force is proportional to displacement, F = k·x, where k is the spring constant in newtons per metre. The energy stored is the area under that force-versus-displacement line — a triangle — which gives PE = ½·k·x². The squared displacement has a big practical consequence: stretching a spring twice as far stores four times the energy, and three times as far stores nine times as much. That is why the last bit of a bow's draw or a trampoline's stretch stores so much more energy than the first bit. Real springs only follow Hooke's law up to their elastic limit; stretch them too far and they deform permanently, so the calculator's ideal-spring result is a close estimate within normal working ranges.
Conservation of energy: turning height into speed
The most powerful use of potential energy is predicting motion without tracking forces over time. When an object falls freely from rest, the principle of conservation of energy says its gravitational PE converts entirely into kinetic energy: m·g·h = ½·m·v². The mass cancels, leaving the famous result v = √(2gh) — every object, heavy or light, reaches the same speed falling the same height in a vacuum, exactly as Galileo argued. The Energy Conservation tab shows this exchange as two crossing curves: PE falling to zero while KE rises to meet the original total. Air resistance and friction siphon off some energy in the real world, so the calculator's frictionless figure is an upper bound on the true impact speed.
Limits and edge cases
A few situations fall outside the simple model. The calculator rejects zero or negative mass, height, and spring constant because they are physically meaningless, and it treats height as a non-negative distance above your reference level. Potential energy itself can be defined as negative if you place the zero above the object, but here the inputs are kept positive for clarity. The constant-g assumption limits the gravitational formula to near-surface problems, and the ideal-spring assumption limits the elastic formula to displacements within the spring's elastic range. For decay chains of energy, rotating systems, or relativistic speeds, these classical formulas give a useful first estimate but a more detailed model is needed.