Potential Energy Calculator | PE=mgh

Potential energy (PE) is stored energy due to position or configuration, a raised mass in a gravitational field, a stretched or compressed spring, or two separated electric charges. Kinetic energy (KE = ½mv²) is energy of motion. Together they constitute mechanical energy. In a conservative system with no friction or energy dissipation, the total E = KE + PE is constant at all times: as an object falls, PE decreases and KE increases by exactly the same amount.

• ›Gravitational PE = mgh: depends only on height above the reference level, not on the path taken to get there
• ›Elastic PE = ½kx²: stored in a spring when compressed or stretched by distance x
• ›Electric PE = k_e q₁q₂/r: energy stored in the configuration of two point charges
• ›KE = ½mv²: energy of a mass in motion, zero when stationary, maximum at the lowest point of a fall
• ›At the highest point of a pendulum swing: all PE, zero KE; at the bottom: all KE, zero PE

The velocity v = √(2gh) is the speed an object reaches at ground level when dropped from rest at height h, ignoring air resistance. It comes directly from energy conservation: mgh = ½mv², so v = √(2gh). Crucially, the mass m cancels, this speed is independent of how heavy the object is. A feather and a hammer, released from the same height in a vacuum, hit the ground at exactly the same speed. Astronaut David Scott famously demonstrated this on the Moon in 1971.

The spring constant k (N/m) measures stiffness: it is the restoring force per unit displacement from equilibrium (Hooke's Law: F = kx). Higher k means a stiffer spring that requires more force to stretch or compress by the same distance. Elastic PE = ½kx² is always positive regardless of direction, the spring stores energy in both compression and extension because only x² appears in the formula.

• ›Soft toy slinky: k ≈ 1–10 N/m, stretches easily under its own weight
• ›Typical pen spring: k ≈ 100–300 N/m
• ›Door return spring: k ≈ 300–800 N/m
• ›Car suspension spring: k ≈ 10,000–50,000 N/m, stiff enough to support the vehicle weight
• ›Atomic bonds (modelled as harmonic oscillators): k ~ 10–1000 N/m depending on bond type

Electric potential V (voltage) at distance r from a point charge q₁ is V = k_e·q₁/r (in volts). The electric PE of a test charge q₂ placed in that field is PE = q₂·V = k_e·q₁q₂/r. This means PE = charge × voltage, making voltage the potential energy per unit charge. The electron-volt (eV) unit is defined directly from this relationship: 1 eV is the PE gained by one electron (q = 1.602×10⁻¹⁹ C) moved through a potential difference of 1 volt.

• ›1 eV = 1.602×10⁻¹⁹ J, the natural energy unit for atomic, molecular, and nuclear processes
• ›Photon of visible green light: ~2.3 eV; X-ray photon: ~10,000 eV = 10 keV
• ›Hydrogen atom ionisation energy: 13.6 eV (energy to remove the electron from the ground state)
• ›Nuclear binding energies: millions of eV (MeV), why nuclear reactions release so much more energy than chemical ones

The sign convention sets PE = 0 when the charges are infinitely far apart. Bringing two opposite-sign charges (q₁q₂ < 0) together releases energy, the system "falls" to a lower energy state spontaneously, so PE becomes negative. This negative PE represents the binding energy: the amount of energy you must supply to pull them apart to infinity. Bringing same-sign charges together requires continuous input of energy (you must do work against the repulsion), so PE becomes positive.

• ›Hydrogen atom: electron−proton PE at the Bohr radius = −27.2 eV (the binding keeps the atom together)
• ›Two electrons 1 nm apart: PE = +1.44 eV (repulsive, positive, work needed to bring them this close)
• ›The magnitude |PE| is the binding energy: how much energy to ionise the system
• ›In chemistry, negative PE = stable bonded configuration; positive PE = unstable, will repel

The joule (J) is the SI unit of energy and should be the default for physics and engineering calculations. Different fields have adopted specialised units that are more convenient at their typical energy scales, using J for atomic energies or kWh for atomic reactions would result in unwieldy numbers. This calculator shows all five unit conversions simultaneously so you can read the result in whichever unit is most familiar for your application.

• ›Joule (J): physics, mechanics, SI standard, use by default
• ›Kilojoule (kJ): thermodynamics, chemistry, food nutrition
• ›Calorie (cal): chemistry lab reactions; note food "Calories" are kcal
• ›Electron-volt (eV): atomic energy levels, photon energies, nuclear physics
• ›Kilowatt-hour (kWh): electrical energy for bills and grid-scale calculations

PE = mgh assumes a uniform gravitational field (constant g), valid near Earth's surface where height changes are small compared to Earth's radius (6371 km). For large height changes, orbital mechanics, or interplanetary travel, use the universal gravitational PE: PE = −G·M·m/r, where G = 6.674×10⁻¹¹ N·m²/kg², M is the planet mass, and r is the distance from the planet centre. The formula mgh is the linear approximation of this expression valid when h ≪ R_Earth.

• ›mgh error < 0.1% for h < 6 km, fine for most engineering applications
• ›mgh error ≈ 1% at h = 64 km, significant for high-altitude balloons and aircraft
• ›Escape velocity: set PE = 0 at infinity → v_esc = √(2GM/r) = 11.2 km/s for Earth
• ›Orbital mechanics: total energy E = KE + PE = −GMm/(2a) where a is the semi-major axis