Kinetic Energy Calculator

Definition and the joule in the SI

The translational kinetic energy of a body of mass m and speed v is Ek = ½mv². It is the energy stored in the motion of the center of mass, not a separate kind of energy. The SI unit is the joule: 1 J = 1 N·m = 1 kg·m²/s². The name honors James Prescott Joule. The unit is defined in the SI Brochure (BIPM, 9th edition).

For 1,000 kg at 20 m/s the steps are short. Step 1: v² = 20² = 400 m²/s². Step 2: m·v² = 1,000·400 = 400,000 kg·m²/s². Step 3: half of that product is 200,000 J. The same mass at 40 m/s has v² = 1,600 and an energy of 800,000 J. A mass of 2,000 kg at 20 m/s has 400,000 J, because mass enters the formula linearly.

History: Huygens, Leibniz, and the factor ½

In De motu corporum ex percussione, written about 1656 and published posthumously in 1703, Christiaan Huygens showed that an elastic collision conserves the sum of mass times speed squared. In “Brevis demonstratio erroris memorabilis Cartesii”, published in Acta Eruditorum in March 1686, Gottfried Wilhelm Leibniz set mv², which he called living force (vis viva), against the Cartesian measure mv. The vis viva dispute ran through the eighteenth century.

Émilie du Châtelet defended mv² in Institutions de physique (1740). She cited the 1722 clay experiments of ’sGravesande, in which the depth of the mark grew with the square of the speed. In Du calcul de l’effet des machines (1829), Gaspard-Gustave Coriolis introduced the factor ½ so that the change in ½mv² would equal the work of a force. William Thomson and Peter Guthrie Tait spread the name kinetic energy for that quantity in Treatise on Natural Philosophy (1867). In 1807 Thomas Young used the word energy for a quantity proportional to mv², still without the factor ½. This page does not invent DOIs, because these works predate that system.

The work–energy theorem

For a particle of constant mass, Newton’s second law integrated along the path gives the work–energy theorem. Net work equals the change in kinetic energy: W = ½mv₂² − ½mv₁². Starting from rest, the work required to reach speed v is exactly ½mv². For 1,000 kg at 20 m/s that is 200,000 J.

The calculator reports that store of energy. It has no force field and no distance field, so it does not turn joules into meters. If a constant force of 2,000 N accelerated that mass from rest to 20 m/s, the distance would be 100 m, because 200,000 J / 2,000 N = 100 m. That illustrates the theorem. It is not an output of this form. Distance at a known deceleration belongs on the decelerated-motion page.

Momentum, the square of speed, and numeric examples

Momentum p = mv is linear in speed and has the unit kg·m/s. The energy can be written Ek = p²/(2m). For 1,000 kg at 20 m/s the momentum is 20,000 kg·m/s, and p²/(2m) = 400,000,000 / 2,000 = 200,000 J, in agreement with the panel. Doubling the speed doubles the momentum and quadruples the energy. Doubling the mass at the same speed doubles both.

A 1,400 kg car at 13.9 m/s, about 50 km/h, has 135,247 J. At 33.3 m/s, about 120 km/h, it has 776,223 J. An 8 g bullet at 350 m/s has 490 J. That is the kinetic energy of the center of mass, without the chemical energy of the charge and without spin about the bullet’s axis. An 80 kg sprinter at 10 m/s has 4,000 J. A 50,000 kg train at 20 m/s has 10 MJ.

The order of magnitude for Earth in orbit about the Sun uses a mass of about 5.972×10²⁴ kg and a mean orbital speed of about 29.78 km/s, or 29,780 m/s, from the NASA planetary fact sheet. Classical ½mv² is then about 2.648×10³³ J. In Newtonian mechanics the total energy of a circular orbit is negative and equal in magnitude to this kinetic energy, because the potential energy is twice as large and negative. The calculator does not compute the gravitational constant, the semimajor axis, or that difference. It also omits the reduced mass, a correction on the order of the Earth-to-Sun mass ratio, about 3×10⁻⁶.

The relativistic limit

Relativistic kinetic energy is (γ − 1)mc², where γ = 1/√(1 − v²/c²) and c = 299,792,458 m/s. The low-speed expansion begins with ½mv². At 20 m/s the relative difference is on the order of 10⁻¹⁵ and does not change any digit of the result. One tenth of the speed of light already leaves the classical value about 0.75% low. Half the speed of light leaves it about 19% low, and at 0.8c the classical result is about 48% of the relativistic kinetic energy.

The calculator always returns ½mv². It does not multiply by γ and it has no relativistic mode. When the entered speed is at least 0.1c, the note beside the result recalls this limit, but the number on the panel stays classical. For Earth’s orbit the relative correction is on the order of 10⁻⁸ and is not computed either.

What this formula does not compute

The formula does not include rotation. A rolling wheel also has ½Iω², and the share of that term depends on the shape of the body. The sum belongs on the rotational kinetic energy page, not here. The formula does not include friction, air resistance, deformation, or the heat of an inelastic collision. It also does not include the chemical energy of a cartridge: the 490 J of the bullet is the energy the bullet already carries, not the energy of the propellant.

The formula is not orbital energy. The Earth example shows the order of magnitude of ½mv² and nothing more. It does not replace celestial mechanics or relativistic dynamics.

Sources

The definition, the unit, and the work–energy theorem follow mechanics textbooks and the historical sources: D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, chapters on work and kinetic energy; H. D. Young and R. A. Freedman, University Physics, the chapter on work and kinetic energy; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume I, chapters 4 and 13; C. Huygens, De motu corporum ex percussione, 1703 edition; G. W. Leibniz, Acta Eruditorum, March 1686; G.-G. Coriolis, Du calcul de l’effet des machines, 1829; W. Thomson and P. G. Tait, Treatise on Natural Philosophy, 1867; BIPM, SI Brochure, 9th edition; NASA, Planetary Fact Sheet (Earth mass and orbital speed). The page is educational.