# Kinetic energy **Kinetic energy** is the energy a body has because it is moving: the [[Work_(physics)|work]] that must be done on it to bring it from rest to its present speed, and the work it can do on something else while coming back to rest. For a body of mass m moving at a speed v far below the [[Speed_of_light|speed of light]], that work is K = ½·m·v², and because work is force times distance, the unit is the newton-metre, which is the [[Joule|joule]].[^murphy-work] The square carries the practical weight: doubling a speed quadruples the [[Energy|energy]] needed to reach it, and quadruples the energy that must be got rid of to stop. In the microsim below the reader sets a launch speed v₀ and picks a braking surface, and a block slides to a stop along a straight track. The stopping distance follows from setting the work done by [[Friction|friction]] equal to the kinetic energy that has to be destroyed: μ·m·g·d = ½·m·v₀², so d = v₀²/(2·μ·g), with the mass cancelling out of both sides. A second preset replaces the brake with a second block, so that [[Momentum|momentum]] is conserved across the impact while kinetic energy is not; the bar that is missing afterwards is the [[Thermal_energy|thermal energy]] the collision made. On the [[Energy]] flagship this article is the child of Part I — Forms, section *Kinetic energy* (row E2), which shares its work-and-energy dial with [[Potential_energy|potential energy]] and [[Mechanical_energy|mechanical energy]], where the same track appears with the brake off and a hill on. ## History and etymology The quantity now called kinetic energy entered physics as a rival to [[Momentum|momentum]] in a dispute over what is conserved when bodies collide. [[Isaac_Newton|Newton]]'s mechanics kept the product of mass and velocity; in 1686 [[Gottfried_Wilhelm_Leibniz|Leibniz]] argued in the *Acta Eruditorum* that the conserved "living force", or [[Vis_viva|vis viva]], was instead the product of mass and the square of speed.[^leibniz1686] The argument ran for decades because both sides were partly right: momentum is a vector conserved in every collision, while m·v² is a scalar conserved only when no energy leaves as heat or sound. [[Émilie_du_Châtelet|Émilie du Châtelet]], in her *Institutions de physique* of 1740, set out the case for vis viva and appealed to Willem 's Gravesande's brass balls dropped into soft clay, where the depth of the impression rises with the square of the impact speed rather than in proportion to it.[^duchatelet1740] The modern word arrived later. [[Thomas_Young_(scientist)|Thomas Young]] used *energy* in something close to its present technical sense in his *Course of Lectures on Natural Philosophy and the Mechanical Arts* of 1807.[^young1807] [[Gaspard-Gustave_de_Coriolis|Gaspard-Gustave de Coriolis]], writing in 1829 on the calculation of the effect of machines, put the factor of one half in front of m·v² so that the quantity would equal the accumulated work, and gave *travail* — work — its technical meaning.[^coriolis1829] The adjective *kinetic*, marking energy of motion off from the energy of position that W. J. M. Rankine named *potential* in 1853, belongs to the same decade, when [[Thermodynamics|thermodynamics]] was forcing physicists to name the stores of energy they were converting into one another.[^rankine1853] ## Overview Energy is a single currency spent in many denominations, and kinetic energy is the denomination of bulk motion. A survey of the forms lists m·g·h for hydro and tidal schemes, h·ν for sunlight, H − T·S for chemical change, c_p·m·ΔT for warmth, q·V for electricity, m·c² for nuclear reactions — and ½·m·v² wherever a mass is already moving, which in practice means [[Wind_power|wind]] and ocean currents as well as vehicles and projectiles.[^murphy-table] The bookkeeping is exact and easy to check by hand. Work is force through distance when the two are aligned, so 2 N through 0.5 m and 0.1 N through 10 m both deliver one joule, and 150 N through 5 m delivers 750 J.[^murphy-work] A falling apple carrying 7 J in total keeps a running ledger as it drops: 7 J of position and none of motion at release, then 5 and 2, then 3 and 4, then 1 and 6 just before it lands, the sum never moving.[^murphy-apple] That ledger is the whole of [[Conservation_of_energy|energy conservation]] for a body under gravity, and the microsim's track is the same ledger laid on its side, with friction rather than height doing the accounting. ## Kinetic energy for non-relativistic velocity The defining statement is the work–energy theorem: the net work done on a body equals the change in its kinetic energy, W_net = ΔK. Applied to a braking block it is a one-line derivation. The [[Friction|kinetic friction]] force is f = μ·N = μ·m·g on level ground, it acts over the stopping distance d, and it must absorb all of ½·m·v₀².[^os-friction] Setting μ·m·g·d = ½·m·v₀² gives d = v₀²/(2·μ·g): the mass cancels, so a loaded lorry and an empty one stop in the same distance on the same surface, and the distance goes as the square of the speed. The microsim makes that square visible. The control is v₀, from 0 to 40 m/s, with a surface selector carrying the tabulated static and kinetic coefficients — rubber on dry concrete 1.0 and 0.7, steel on steel 0.6 and 0.3, ice on ice 0.1 and 0.03.[^os-friction] A 1,200 kg car at 14 m/s carries 118 kJ and needs 14.3 m of dry concrete; at 28 m/s it carries 470 kJ and needs 57.1 m, four times as far for twice the speed (derived). On ice at μ = 0.03 the same 14 m/s needs 333 m (derived), which is why the sim draws the distance to scale against a row of parked cars rather than as a number. The model is ILLUSTRATIVE: it holds μ constant, ignores [[Drag_(physics)|air drag]] and reaction time, and treats braking as a perfectly locked slide, so it is a floor on real stopping distance, not a prediction of one. ### Kinetic energy of rigid bodies A rigid body carries kinetic energy in two independent accounts. Its translational store, ½·M·v_G², depends only on the speed of the centre of mass and treats the body as a point of mass M; its rotational store depends on how that mass is arranged about the spin axis. The two simply add: K = ½·M·v_G² + ½·I·ω², with no coupling term, which is why a spinning skater gliding across a rink can be analysed as a moving point and a rotating shape at once.[^idema-rot] For a body that is not rotating the second term vanishes and the familiar ½·m·v² is recovered. ### Rotating bodies The rotational store uses the [[Moment_of_inertia|moment of inertia]] I = Σ m·r², the mass-weighted sum of squared distances from the axis, in the place mass takes in the translational store, with angular speed ω in the place of v: K_rot = ½·I·ω².[^idema-rot] Standard shapes give I as a pure number times M·R²: a thin hoop 1, a solid cylinder ½, a hollow sphere ⅔, a solid sphere ⅖, a rod about its centre (1/12)·M·L² and about its end (1/3)·M·L².[^idema-rot] Shifting the axis away from the centre of mass by a distance d adds M·d², which is the parallel-axis theorem. The rotational store is not a minor correction. A helicopter with four 4.00 m, 50.0 kg blades turning at 300 rpm while the 1,000 kg aircraft flies at 20.0 m/s has a translational store of 2.00×10⁵ J against a rotational store of 5.26×10⁵ J, a ratio of 0.380 — most of the machine's kinetic energy is in the rotor, not in the flight.[^os-heli] That is the principle of the [[Flywheel|flywheel]] and of [[Flywheel_energy_storage|flywheel energy storage]], where the point is to put energy into ω rather than v. A body [[Rolling|rolling]] without slipping splits its store in the fixed ratio K_rot/K = β/(1 + β), with β = I/(M·R²), so a solid cylinder keeps a third of its energy in spin and a hoop keeps half (derived).[^idema-roll] ### Kinetic energy of systems For a collection of particles the kinetic energy separates into the energy of the whole and the energy within: K = ½·M·v_G² + K_internal, where v_G is the velocity of the centre of mass computed from the system's total [[Momentum|momentum]].[^richards-momentum] The split matters because the two halves obey different rules in a collision. Momentum accounting treats an impact as a transfer that leaves the system total untouched, so v_G is the same before and after; kinetic energy is free to fall, and collisions are classified by what it does — unchanged in an elastic collision, reduced in an inelastic one, reduced to the momentum-limited minimum when the bodies stick, and increased only when a store such as a spring or a charge is released.[^os-collisions] The microsim's collision preset uses a textbook pair of pucks: 12 g moving at 2.5 m/s strikes 15 g at rest, and in the elastic case the lighter puck rebounds at 0.278 m/s while the heavier moves off at 2.22 m/s, with the total 37.5 mJ intact (derived).[^os-pucks] Dragging the restitution control down to zero makes the pucks stick, and the kinetic-energy bar drops while the momentum bar does not move. Real crashes sit near that end. A 1,200 kg car striking a 3,000 kg truck hard enough to push it 10 m across a surface of μ = 0.62 must have been travelling at 27.6 m/s, the truck leaves at 11.0 m/s, and about 60 % of the original kinetic energy is gone into deformation and heat (derived).[^os-crash] ### Fluid dynamics A moving fluid has no single velocity, so its kinetic energy is counted per unit volume as ½·ρ·v², the quantity that appears in [[Bernoulli's_principle|Bernoulli's principle]] as dynamic pressure and trades against static pressure along a streamline.[^richards-momentum] Multiplying by the volume crossing an area each second turns it into the power in a stream: P = ½·ρ·A·v³, cubic rather than quadratic because faster air also delivers more mass per second. That cube governs [[Wind_power|wind power]], and no turbine can take all of it, since the air must keep enough speed to leave — the Betz limit caps the fraction near 0.59.[^kerlin-wind] In [[Turbulence|turbulent]] flow the same ½·ρ·v² spreads across eddies of many sizes and is drained to heat by [[Viscosity|viscosity]] at the smallest, which is why [[Fluid_mechanics|fluid mechanics]] treats turbulent kinetic energy as a budget with production and dissipation terms. ### Frame of reference Kinetic energy is not a property of a body alone but of a body and an observer. A forklift driving at 3 m/s along a flatcar that is itself moving at 5 m/s is doing 8 m/s to the trackside, or 2 m/s if it drives the other way, and its kinetic energy differs by a factor of sixteen between those cases although nothing about the forklift changed.[^richards-momentum] There is no privileged frame in which the "true" value is found. What survives is the physics: the work–energy theorem holds in every inertial frame, because the work done changes with the frame too and the two changes cancel. The internal term K_internal above is the part every inertial observer agrees on, which is why a collision's energy loss is frame-independent even though its kinetic energy is not. ### Rotation in systems When a body both spins and translates, a constraint can tie the two stores together. Rolling without slipping imposes v = ω·R, so β fixes the split once and for all and a rolling body accelerates down a slope at g·sin θ/(1 + β) — a solid cylinder reaches (2/3)·g·sin θ, slower than a sliding block of any mass.[^idema-roll] Where the constraint is not yet satisfied, friction enforces it and charges for the service: a cylinder launched at v₀ with no spin skids until friction has spun it up, ends rolling at v₀/(1 + β) however rough the surface, and leaves the fraction β/(1 + β) of its kinetic energy — a third for a solid cylinder — as heat in the skid mark (derived).[^idema-roll] ## Relativistic kinetic energy At speeds approaching that of light the Newtonian expression fails, and [[Special_relativity|special relativity]] replaces it with K = (γ − 1)·m·c², where γ = 1/√(1 − v²/c²) is the Lorentz factor. The total energy is E = γ·m·c², of which m·c² remains at rest, so kinetic energy becomes the excess of total energy over rest energy rather than a separate quantity.[^os-collisions] Because γ diverges as v approaches c, reaching the [[Speed_of_light|speed of light]] would take infinite work and no massive body can be brought to it. At v = 0.866·c, γ equals 2 and the kinetic energy equals the rest energy (derived) — the point where an accelerator has spent as much on motion as the particle's own [[Mass–energy_equivalence|mass–energy]]. ### Low speed limit Expanding the Lorentz factor for small v/c gives K = ½·m·v² + (3/8)·m·v⁴/c² + …, so the Newtonian formula is the leading term of a series rather than a different law. The first correction is smaller by (3/4)·(v/c)², so at 0.1·c the Newtonian value is low by about 0.75 %, and at orbital speeds by about one part in 10¹² (derived). That is why the braking block in the microsim, and every vehicle and projectile on Earth, can be handled with ½·m·v² unqualified, and why the relativistic form is the working formula only in accelerators, cosmic-ray physics and [[Nuclear_fission|nuclear]] reaction accounting. ### General relativity In [[General_relativity|general relativity]] energy of every kind, kinetic energy included, enters the stress–energy tensor and so contributes to the curvature of spacetime: a hot, fast-moving gas gravitates slightly more strongly than the same gas at rest. The price is that the clean split between kinetic and potential energy, and the global conservation law itself, is defined only in spacetimes with the right symmetry. A static gravitational field has a conserved energy for each orbiting body and the Newtonian ledger survives as an approximation; a general expanding spacetime has no such quantity, and [[Conservation_of_energy|conservation of energy]] holds locally rather than globally. ## Kinetic energy in quantum mechanics In [[Quantum_mechanics|quantum mechanics]] kinetic energy is an operator rather than a number. Momentum is represented by −iħ·∂/∂x, so kinetic energy is T̂ = −(ħ²/2m)·∇², and the [[Schrödinger_equation|Schrödinger equation]] iħ·∂Ψ/∂t = −(ħ²/2m)·∂²Ψ/∂x² + U·Ψ is the statement that the total energy is the sum of that operator and the potential.[^likharev-qm] What can be measured is the expectation value ⟨T⟩ = −(ħ²/2m)·∫Ψ*·∇²Ψ·dx, which is large where the [[Wave_function|wave function]] is sharply curved: confining a particle more tightly forces more curvature into it and so costs kinetic energy. That cost has no classical counterpart. A particle confined to a box of length L has E_n = n²·h²/(8·m·L²), entirely kinetic, and the lowest state is not zero — the [[Zero-point_energy|zero-point energy]] that remains at absolute zero and that the [[Uncertainty_principle|uncertainty principle]] requires, since a particle of zero position spread would have unbounded momentum spread.[^likharev-qm] The same accounting sets the quantum harmonic oscillator's ground state at ħ·ω₀/2 and supplies the kinetic term that balances electrostatic attraction in an [[Atom|atom]]: without it the [[Electron|electron]] would fall into the nucleus.[^likharev-qm] The classical link is restored on average, because a gas in [[Statistical_mechanics|thermal equilibrium]] has a mean translational kinetic energy of (3/2)·k_B·T per molecule whichever mechanics derives it.[^os-kinetic-theory] ## See also - [[Rotational_energy]] - [[Vis_viva]] - [[Potential_energy]] - [[Mechanical_energy]] - [[Work_(physics)]] - [[Momentum]] - [[Inelastic_collision]] ## Notes Explanatory material is carried in the body rather than in separate notes; every footnote definition on this page, bibliographic and explanatory alike, is collected under References below. Values marked "(derived)" were computed for this article from the inputs the cited source prints, not read off the source's own printed answer. ## References [^murphy-work]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 5 "Energy and Fossil Fuels", pp. 88–89 (W = F·d "when the motion is aligned with the direction of force"; 1 J = 1 N·m; the 2 N × 0.5 m, 0.1 N × 10 m and 150 N × 5 m examples). Portal Book 097, https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-apple]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 5, pp. 90–91 (the falling-apple ledger, total 7 J, stepping through (7, 0), (5, 2), (3, 4) and (1, 6) J). Portal Book 097. [^murphy-table]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 5, p. 90, Table 5.2 (the forms of energy and where each reappears: m·g·h, ½·m·v², h·ν, H − T·S, c_p·m·ΔT, q·V, m·c²). Portal Book 097. [^leibniz1686]: Leibniz, G. W. (1686). "Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturalem." *Acta Eruditorum*, March 1686. [^duchatelet1740]: du Châtelet, Émilie (1740). *Institutions de physique*. Paris: Prault fils. (Her exposition of vis viva and of 's Gravesande's clay-impression experiments.) [^young1807]: Young, Thomas (1807). *A Course of Lectures on Natural Philosophy and the Mechanical Arts*. London: Joseph Johnson. [^coriolis1829]: Coriolis, Gaspard-Gustave de (1829). *Du calcul de l'effet des machines*. Paris: Carilian-Gœury. [^rankine1853]: Rankine, W. J. M. (1853). "On the General Law of the Transformation of Energy." *Proceedings of the Philosophical Society of Glasgow*. [^os-friction]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 1*. OpenStax. Chapter 6, pp. 278–279 (f_s ≤ μ_s·N, f_k = μ_k·N, and Table 6.1: rubber on dry concrete 1.0/0.7, steel on steel 0.6/0.3, ice on ice 0.1/0.03). Portal Book 077, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^os-collisions]: Sanny and Ling (2016), *University Physics Volume 1*, Chapter 9 "Linear Momentum and Collisions", pp. 417–418 (collisions classified by what happens to kinetic energy: explosion, inelastic, perfectly inelastic, elastic). Portal Book 077. [^os-pucks]: Sanny and Ling (2016), *University Physics Volume 1*, Chapter 9, Example 9.11, pp. 419–420 (12 g at 2.5 m/s strikes 15 g at rest; the book states that the lighter puck reverses, and the outcome velocities −0.278 m/s and +2.22 m/s are derived from its inputs). Portal Book 077. [^os-crash]: Sanny and Ling (2016), *University Physics Volume 1*, Chapter 9, Example 9.13, pp. 421–423 (1,200 kg car into a 3,000 kg truck that slides 10 m at μ = 0.62; the truck's 11.0 m/s, the car's 27.6 m/s and the 60 % kinetic-energy loss are derived from the book's inputs). Portal Book 077. [^os-heli]: Sanny and Ling (2016), *University Physics Volume 1*, Chapter 10, Example 10.9, pp. 487–488 (four 4.00 m, 50.0 kg blades at 300 rpm on a 1,000 kg aircraft at 20.0 m/s; K_trans/K_rot = 0.380). Portal Book 077. [^idema-rot]: Idema, Timon (2018). *Mechanics and Relativity*. Chapter 5, pp. 66–67 (I = Σ m·r²; Table 5.1 of standard moments of inertia; the parallel-axis theorem; K_rot = ½·I·ω²). Portal Book 080, https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity [^idema-roll]: Idema (2018), *Mechanics and Relativity*, Chapter 5, pp. 68–71 (the rolling condition v = ω·R; the solid cylinder's a = (2/3)·g·sin θ derived three ways; skid-to-roll at v_r = v₀/(1 + I/(m·R²)), independent of μ_k). The energy fractions β/(1 + β) are derived. Portal Book 080. [^richards-momentum]: Richards, Donald (2001). *Basic Engineering Science: A Systems, Accounting, and Modeling Approach*. Chapter on linear momentum, pp. 145–166 (P = m·V; P_sys = Σ m_j·V_j and V_G = P_sys/m_sys; relative velocity V_B = V_A + V_B/A and the forklift-on-a-flatcar example at p. 151; velocities must be measured in an inertial frame). Portal Book 020, https://open.umn.edu/opentextbooks/textbooks/basic-engineering-science-a-systems-accounting-and-modeling-approach [^kerlin-wind]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*. Chapter 5 "Wind", pp. 217–231 (P = ½·ρ·A·v³, the cubic law and the Betz cap of about 0.59, with Tables 8-1 and 8-2). Portal Book 048, https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges [^likharev-qm]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2 "1D wave mechanics", p. 31 (iħ·∂Ψ/∂t = −(ħ²/2m)·∂²Ψ/∂x² + U·Ψ; normalization of ΨΨ*) and pp. 32–33 (⟨Δx²⟩⟨Δp²⟩ ≥ ħ²/4); the harmonic-oscillator ladder E_n = ħ·ω₀·(n + ½) is in the same chapter (page to pin). Portal Book 047, https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^os-kinetic-theory]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*. OpenStax. Chapter 2 "The Kinetic Theory of Gases", p. 88 (KE_avg = (3/2)·k_B·T). Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 ## External links - [University Physics Volume 1](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1), OpenStax — collisions, friction and moment of inertia, Portal Book 077 - [Mechanics and Relativity](https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity), Timon Idema — the rotation chapter, Portal Book 080 - [Energy and Human Ambitions on a Finite Planet](https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet), Thomas Murphy — Portal Book 097 - The Wikipedia pair's external links list further open resources <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Kinetic_energy.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Kinetic energy* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Kinetic_energy.html" data-title="Kinetic energy"></div> *Built from `MICROSIM_GUIDE/specs/sims/Kinetic_energy.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Kinetic_energy) : [Wikitube](https://en.wikitube.io/wiki/Kinetic_energy) · pinned revision [1370969006](https://en.wikipedia.org/w/index.php?oldid=1370969006) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E2 · sim pending (matter/Kinetic_energy).*