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Calculate gravitational potential energy (PE = mgh), find height from potential energy, or determine mass from PE and height. Also calculates elastic potential energy using Hooke's Law (PE = ½kx²).
Potential energy is stored energy — energy an object has due to its position or configuration. Gravitational PE = mgh depends on mass, gravity, and height above a reference point. Elastic PE = ½kx² is stored in compressed or stretched springs. Both convert to kinetic energy when released.
Potential energy (PE) is energy stored in an object due to its position, arrangement, or condition — ready to be converted to kinetic energy or do work. The two most common forms in classical physics are gravitational potential energy (due to height above a reference) and elastic potential energy (due to deformation of a spring or elastic material).
The law of conservation of energy states that in a closed system, total mechanical energy (KE + PE) remains constant. A ball dropped from height h: at the top, all energy is PE = mgh. As it falls, PE converts to KE. Just before hitting the ground: all energy is KE = ½mv² = mgh, giving v = √(2gh). This relationship between height and impact velocity is used in everything from roller coaster design to ballistics.
| Application | Reference Point (PE=0) | Notes |
|---|---|---|
| Everyday objects (tables, shelves) | Floor level or ground | Heights measured from floor |
| Hydroelectric dam water | Sea level or turbine level | Height = dam height, typically 30-300m |
| Aircraft fuel burn (altitude) | Sea level | Height = cruising altitude ~10,000m |
| Roller coaster | Lowest point of track | Height = drop from peak |
| Planetary orbit | Infinite (convention) | PE = 0 at infinite distance, negative closer |
Roller coaster: starting at height h = 40m above ground (bottom of first drop). PE = mgh = 70kg × 9.81 × 40 = 27,468 J. At bottom: all converted to KE. v = √(2gh) = √(2 × 9.81 × 40) = 28.0 m/s = 100.7 km/h. Real coasters achieve ~90-95% of this (friction/air resistance losses). Pendulum: at max height h, all PE. At bottom, all KE. Period T = 2π√(L/g) — independent of mass and amplitude for small swings. Hydroelectric dam: water at height h = 100m. PE per kg = mgh = 1×9.81×100 = 981 J/kg. Flow rate 500 m³/s = 500,000 kg/s. Power = 500,000 × 981 = 490.5 MW (theoretical, practical ~85% = 417 MW).
Hooke's Law: F = -kx (spring force = spring constant × displacement, restoring direction). Elastic PE = ½kx². Spring constant k: rubber band ~100 N/m, car suspension spring ~20,000 N/m, stiff industrial spring ~100,000 N/m. A car suspension spring (k = 25,000 N/m) compressed 0.1m: PE = ½ × 25,000 × 0.1² = 125 J. This energy is released to restore the car body as the wheel rebounds. Simple harmonic motion: mass on spring oscillates with period T = 2π√(m/k). Energy alternates between KE (at equilibrium) and PE (at maximum displacement) — total conserved.
Potential energy extends beyond mechanical systems. Chemical PE: stored in molecular bonds. Combustion releases chemical PE as heat and light. Gasoline: ~46 MJ/kg energy density. TNT: ~4.6 MJ/kg. Batteries: lithium-ion ~0.7 MJ/kg. Nuclear PE: stored in atomic nuclei. Fission of uranium releases ~80 million times more energy per kg than chemical explosion. Nuclear PE arises from the strong nuclear force binding protons and neutrons. Mass-energy equivalence: E = mc². Even 1g of mass contains c² × 0.001 = 9×10¹³ J — equivalent to Hiroshima bomb energy.
Always ensure units are consistent before calculating. This calculator uses SI units (International System): meters (m) for distance, kilograms (kg) for mass, seconds (s) for time, Newtons (N) for force, Joules (J) for energy, Watts (W) for power, and Pascals (Pa) for pressure. When converting from other unit systems (imperial, CGS), convert all values before entering them into the calculator.
Gravitational acceleration on Earth: g = 9.81 m/s² (standard: 9.80665 m/s²). Speed of light in vacuum: c = 2.998×10⁸ m/s. Universal gravitational constant: G = 6.674×10⁻¹¹ N·m²/kg². Speed of sound in air at 20°C: 343 m/s. Atmospheric pressure at sea level: 101,325 Pa = 1 atm. These constants appear in virtually every branch of physics and engineering.
Physics calculations directly inform engineering safety standards. Stopping distance calculations (kinetic energy / braking force) set speed limits and vehicle safety requirements. Pressure calculations determine pipe and vessel wall thickness in plumbing and industrial equipment. Torque specifications on fasteners prevent structural failures. Centripetal acceleration limits guide road curve design and roller coaster safety. Every safety standard in transportation, construction, and consumer products has physics calculations at its foundation.