Potential Energy Calculator

Computes gravitational potential energy from PE = mgh, where m is mass, g is gravitational acceleration (9.80665 m/s² standard on Earth), and h is height above a reference point.

The formula assumes uniform gravity, valid for height changes small relative to Earth's radius. For satellite orbits or interplanetary calculations, the more general −GMm/r form is needed.

Gravitational potential energy in a uniform field is the work required to lift mass m through height h against gravitational acceleration g. The formula PE = mgh is a linearization of the full gravitational potential energy U = −GMm/r near the surface of a large body, where r changes are small compared to the body's radius. The minus sign in the general form reflects the convention that potential energy is zero at infinite separation; near Earth's surface this convention is replaced by the more practical 'zero at ground level' choice, with the sign absorbed into the sign convention for h.

A worked example: lifting a 70 kg person up a 10-meter staircase requires PE = 70 × 9.80665 × 10 ≈ 6,865 joules, or about 1.64 kcal of metabolic energy at perfect efficiency. Real human efficiency is roughly 25%, so the actual metabolic cost is closer to 6.5 kcal. A 1 kg book on a shelf 2 meters above the floor stores 19.6 J relative to the floor; if it falls and hits the floor, those 19.6 J convert to kinetic energy, giving an impact speed of v = √(2×9.80665×2) ≈ 6.26 m/s.

Limitations and the choice of reference point matter. PE is always relative; only differences are physically meaningful. The reference height h = 0 can be set to any convenient point: ground level, sea level, the bottom of a well, the center of mass of a starting configuration. As long as the same reference is used throughout a problem, the chosen origin doesn't affect the physics. Beyond Earth's surface, g varies: at 100 km altitude, g has decreased by about 3%; in low Earth orbit (~400 km), about 12%. For satellite trajectories the full −GMm/r form is required.

The formula extends naturally to other bodies with different g values: Moon's g ≈ 1.62 m/s² (about 1/6 Earth's), Mars's g ≈ 3.71 m/s². A 1 kg mass lifted 1 meter on the Moon stores about 1.62 J versus 9.81 J on Earth, which is why astronauts could leap so easily during Apollo missions. For aerospace applications, the more general gravitational potential combined with kinetic energy gives the orbital mechanics relationships (vis-viva equation, escape velocity), all of which reduce to mgh in the appropriate limit. The conservation of mechanical energy, KE + PE = constant in the absence of dissipative forces, underlies pendulum motion, projectile trajectories, and the relationship between drop height and impact velocity.