sqrt2gh

sqrt(2 g h) denotes the speed of a body released from rest and falling vertically a distance h in a uniform gravitational field g, under the simplifying assumption of negligible air resistance. It is the magnitude of the velocity after the fall, with v = sqrt(2 g h).

Derivation can follow from kinematics or energy principles. From kinematics, v^2 = u^2 + 2 a s, with initial speed u = 0, acceleration a = g, and displacement s = h, giving v^2 = 2 g h. From energy conservation, gravitational potential energy m g h converts to kinetic energy (1/2) m v^2, yielding v = sqrt(2 g h).

Earthly values illustrate the scale. With g ≈ 9.81 m/s^2, a fall of h = 10 m yields v ≈ sqrt(196.2) ≈ 14 m/s. For h = 50 m, v ≈ sqrt(2 × 9.81 × 50) ≈ sqrt(981) ≈ 31 m/s. These numbers reflect the idealized case of no air resistance.

Limitations and scope are important. In real conditions, drag forces slow the acceleration, and objects may approach a terminal velocity that is less than sqrt(2 g h). Air density, object shape, and mass influence this effect. The relation also assumes a vertical fall from rest and a constant gravitational field; deviations occur for non-vertical motion, significant altitude changes, or varying gravity.

See also: free fall, kinematics, energy conservation, v^2 = 2 g h.