# Momentum **Momentum** is the product of a body's mass and its [[Velocity|velocity]], `P = m·V`, a vector pointing along the motion and measured in kilogram-metres per second.[^richards-p] It is the quantity that bookkeeping treats as conserved: the momentum of an isolated system cannot change, whatever its parts do to one another. That makes it more robust than [[Kinetic_energy|kinetic energy]], which the same collision is free to destroy, and it is why collisions, recoil, thrust and [[Pressure|pressure]] are all analysed in the same currency. In the microsim below the reader slides a restitution coefficient e from 0 to 1 and sets the mass ratio of two pucks on a frictionless track. Two equations run the collision: conservation, `m1·v1 + m2·v2 = m1·v1' + m2·v2'`, and the restitution rule, `v2' − v1' = −e·(v2 − v1)`, which between them fix both outgoing speeds.[^os-collisions] A bar for the total momentum and a bar for the total kinetic energy stand side by side. As e falls the momentum bar does not move at all, while the energy bar sinks to a floor at e = 0, where the pucks travel together at the velocity of their common centre of mass. A preset loads a textbook pair — 12 g at 2.5 m/s striking 15 g at rest, which at e = 1 leaves them at −0.278 and +2.22 m/s (derived) — and a second preset lines up five equal balls as a [[Newton's_cradle|Newton's cradle]].[^os-pucks] On the [[Physics]] flagship this page serves Part II — Core theories, in the section *Momentum and collisions* (row P15), where it shares its collision track with [[Newton's_laws_of_motion|Newton's laws of motion]] upstream and with [[Elastic_collision|elastic]] and [[Inelastic_collision|inelastic collisions]] downstream, each of which reuses the same two pucks with e pinned at an end of its range. ## Classical Classical momentum is defined for a point mass and then extended, unchanged in meaning, to systems of particles, continuous bodies and flows — one accounting identity applied at successively larger scales. ### Single particle For one body, `P = m·V`: mass is a scalar, velocity is a vector, and momentum inherits the direction of the motion.[^richards-p] The velocity must be measured in an inertial frame, a condition easy to state and easy to violate — velocities read off a moving platform or the deck of a ship are not admissible until they have been referred back to one.[^richards-frames] Momentum is not a measure of "how hard a body is to stop" in any absolute sense; it measures the impulse — force multiplied by the time it acts — already delivered to the body, and the impulse needed to bring it back to rest. ### Many particles A system's momentum is the sum over its parts, `P_sys = sum(m_j·V_j)`, and dividing by the total mass defines the velocity of the centre of mass, `V_G = P_sys/m_sys`.[^richards-sys] Three particles of 15, 10 and 5 kg moving at (5, 5), (3, −5) and (2, −2) m/s give `P_sys` = 115**i** + 15**j** kg·m/s, so the 30 kg system drifts at (3.83, 0.50) m/s however violently its members interact (derived).[^richards-sys] The internal forces cancel in pairs, by Newton's third law, and drop out of the sum. This is the reason a system can be replaced by a single point of mass `m_sys` at `V_G` for every purpose except the internal accounting, and the reason a [[Inertia|body]] that explodes in flight leaves its fragments' centre of mass on the original trajectory. ### Relation to force Differentiating the sum gives the rate form of the accounting, which is more general than `F = m·a`. For a system that also exchanges mass with its surroundings, `dP_sys/dt = sum(F_ext) + sum_in(mdot·V) − sum_out(mdot·V)`: momentum changes because external [[Force|forces]] act, or because mass carrying momentum crosses the boundary.[^richards-rate] Newton's three laws are special cases of this one statement — the first for a system with no external force, the second for a closed system of constant mass, the third for the internal pairs that cancel.[^richards-rate] Written this way the law applies unchanged to a [[Control_volume|control volume]] with flow through it, which is how a [[Jet_engine|jet engine]] is analysed. ### Conservation If the external forces sum to zero, `P_sys` is constant. Conservation is a law, not a modelling assumption, and the accounting framework treats it as an identity that must balance rather than a result to be checked.[^richards-rate] Its strength is that it says nothing about the mechanism: the [[Force|contact forces]] during a crash are enormous, brief and unmeasurable, and momentum conservation steps over them, relating the state before to the state after. The microsim's momentum bar is that claim made visible — whatever the reader does to e or to the mass ratio, its length is fixed by the initial conditions alone, and only the split between the two pucks changes. ### Dependence on reference frame Momentum is frame-dependent in a way that is easy to mistake for a defect. 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; relative velocities add as vectors, `V_B = V_A + V_B/A`.[^richards-frames] Its momentum therefore differs by a factor of four between those two cases although nothing about the forklift changed. What is not frame-dependent is the conservation law: if `P_sys` is constant in one inertial frame it is constant in every other, because the frames differ by a constant velocity times a constant total mass. The centre-of-mass frame, in which `P_sys` = 0, is the one in which a collision is easiest to read. ### Application to collisions Collisions are classified by what happens to kinetic energy, not to momentum: an explosion increases it, an inelastic collision reduces it, a perfectly inelastic collision reduces it to the minimum that momentum allows, and an [[Elastic_collision|elastic collision]] leaves it alone.[^os-collisions] The microsim's restitution slider sweeps that classification continuously. With the preset masses, the pucks carry 37.5 mJ before the impact; at e = 0 they leave together at `V_G` = 1.111 m/s with 16.7 mJ, so 55.6 % of the energy is gone — a fraction equal to `m2/(m1 + m2)`, independent of speed (derived). At intermediate e the loss scales as `1 − e^2`, so e = 0.5 already destroys 41.7 % of the energy (derived). Two textbook problems show why the momentum equation is used first. Thor swings a hammer through 10 m in 1 s against a 200 kg train that then moves 2 m in 0.75 s; momentum alone fixes the hammer at about 73 kg and the energy lost at about 2.7 kJ (derived).[^os-thor] A 1,200 kg car strikes a 3,000 kg truck hard enough to push it 10 m across a surface of μ = 0.62, which by [[Friction|friction]] work gives the truck 11.0 m/s and, through momentum, the car 27.6 m/s before impact, with about 60 % of the kinetic energy lost (derived).[^os-crash] Two traps recur. The elastic-collision equations are quadratic and carry an unphysical root, the "no collision" solution in which the bodies pass through each other untouched.[^os-collisions] And when friction acts, momentum is conserved only across the impact instant, not across the slide that follows it.[^os-crash] ### Multiple dimensions In two or three dimensions the conservation law is one equation per axis, and the axes do not talk to each other. A glancing collision is solved by resolving both incoming velocities into components and conserving each separately; the third equation needed comes either from the restitution rule along the line of centres or from the elastic energy condition. The microsim's slider demonstrates the head-on case; the same model run with an offset impact parameter gives the familiar result that two equal masses colliding elastically leave at right angles to one another. ### Objects of variable mass A rocket, a leaking hopper and a chain lifted off the floor all change mass while they move, and `F = m·a` fails for them. The rate form does not: the flux terms `sum_in(mdot·V)` and `sum_out(mdot·V)` carry the momentum that arrives or departs with the mass.[^richards-rate] The subtlety is that the flux term uses the velocity of the crossing mass in the chosen inertial frame, not its velocity relative to the body — the source of most sign errors in [[Rocket_propellant|rocket]] problems. The same terms over a fixed [[Control_volume|control volume]] give the thrust of a [[Turbojet|turbojet]] and the reaction force on a pipe bend. ## Generalized Beyond Newtonian point mechanics, momentum is defined by its role rather than by `m·V`, and that generalization is what survives into field theory and [[Quantum_mechanics|quantum mechanics]]. ### Lagrangian mechanics In [[Lagrangian_mechanics|Lagrangian mechanics]] each generalized coordinate `q_i` has a conjugate momentum `p_i = dL/d(qdot_i)`, the derivative of the Lagrangian with respect to the corresponding generalized velocity.[^cline-lagrangian] For a free particle in Cartesian coordinates this reproduces `m·v` exactly; for an angle it produces [[Angular_momentum|angular momentum]]; for a charged particle in a [[Magnetic_field|magnetic field]] it produces the canonical momentum, which differs from `m·v` by a term in the vector potential. The payoff is mechanical: if a coordinate does not appear in L, its conjugate momentum is a constant of the motion, and the problem loses a degree of freedom without any further work. ### Hamiltonian mechanics [[Hamiltonian_mechanics|Hamiltonian mechanics]] promotes the conjugate momenta to independent variables, so a system of N coordinates becomes a point in a 2N-dimensional phase space evolving under `qdot_i = dH/dp_i` and `pdot_i = −dH/dq_i`.[^cline-lagrangian] The symmetry of the two equations is why this formulation, not Newton's, carries over to statistical and quantum mechanics: the [[Schrödinger_equation|Schrödinger equation]] is built by replacing the Hamiltonian's momenta with operators. ### Symmetry and conservation The deepest statement about momentum is that its conservation is not an independent law. Emmy Noether proved in 1918 that every continuous symmetry of a system's action corresponds to a conserved quantity; invariance under translation in space yields conservation of linear momentum, invariance under rotation yields [[Angular_momentum|angular momentum]], and invariance under translation in time yields [[Conservation_of_energy|energy]].[^noether1918] [[Noether's_theorem|Noether's theorem]] therefore says that momentum is conserved because empty space has no preferred place — and, read the other way, that a system pinned to a fixed external structure has no momentum conservation to appeal to, which is why a ball bouncing off a wall conserves momentum only when the planet is included in the accounting. ## Momentum density A continuous medium has no single velocity, so momentum is carried as a density, `m·V` per unit volume, and conservation becomes local: the momentum in a small region changes only by flow across its surface or by force on it. ### In deformable bodies and fluids For a fluid of density ρ moving at velocity **v**, the momentum density is ρ**v**, and its balance over a [[Control_volume|control volume]] is the integral form of the momentum equation; taking the volume to zero gives the [[Navier–Stokes_equations|Navier–Stokes equations]], in which pressure gradients and [[Viscosity|viscous]] stresses play the part of the external forces.[^richards-rate] Momentum transport is what makes viscosity a transport coefficient at all: a fast layer drags a slow one because [[Molecule|molecules]] crossing between them carry momentum with them, and the same picture in a gas gives viscosity from the [[Kinetic_theory_of_gases|kinetic theory of gases]], whose own microsim runs the molecular version of this exchange. In [[Solid_mechanics|solid mechanics]] the same density appears on the left of the equation of motion, with the stress tensor supplying the surface term. ### In electromagnetics Fields carry momentum too. John Henry Poynting showed in 1884 that an electromagnetic field transports energy at a rate given by the vector now named for him, and the same field carries a momentum density proportional to it.[^poynting1884] The consequence is measurable: light presses on what absorbs or reflects it, so [[Sunlight|sunlight]] pushes on a spacecraft and a [[Photon|photon]] of energy E carries momentum E/c. Momentum conservation for charges alone fails — two charges interacting through a field can appear to violate Newton's third law at an instant — and is restored only when the field's own momentum is counted. ## Non-classical ### Quantum mechanical In quantum mechanics momentum is an operator, not a number: `p = −i·hbar·d/dx`, acting on the [[Wave_function|wave function]], and the [[Schrödinger_equation|Schrödinger equation]] `i·hbar·dPsi/dt = −(hbar^2/2m)·d2Psi/dx2 + U·Psi` is the statement that energy is kinetic plus potential with that substitution made.[^likharev-qm] A state of definite momentum is a plane wave spread over all space, so position and momentum cannot both be sharp: `<dx^2>·<dp^2> >= hbar^2/4`.[^likharev-qm] A localized [[Wave_packet|wave packet]] built from a band of momenta of width `dk = 1/(2·dx)` travels at the [[Group_velocity|group velocity]] `v_gr = hbar·k0/m`, which is exactly the classical `p/m`, and spreads as it goes.[^likharev-qm] ### Relativistic At speeds approaching c the Newtonian definition fails and momentum becomes `p = gamma·m·v`, with `gamma = 1/sqrt(1 − v^2/c^2)`; energy and momentum join into the [[Four-momentum|four-momentum]], whose invariant length gives `E^2 = (p·c)^2 + (m·c^2)^2`.[^up3-relativity] The relation holds for massless particles as well, where it reduces to `E = p·c`, which is how a photon carries momentum without carrying mass. Conservation survives the change intact — it is still the total four-momentum that is fixed in a [[Special_relativity|relativistic]] collision — and it is how particle physics infers particles it cannot see. ## History of the concept ### Impetus Before momentum there was *impetus*: the medieval proposal that a projectile carries an impressed motive power which keeps it going and is gradually used up, advanced most influentially by Jean Buridan at Paris in the fourteenth century, who applied it both to thrown stones and to the motion of the heavens.[^buridan] It improved on the Aristotelian account, in which the surrounding air did the pushing, and it contains the germ of the right idea — a quantity carried by the moving body, proportional to its matter and its speed. What it lacked was the recognition that nothing needs to be used up at all. ### Quantity of motion The seventeenth century turned impetus into an additive, conserved quantity through the study of impact. René Descartes held that God conserved the total "quantity of motion" of the universe, but defined it without direction, so his collision rules were wrong. [[Christiaan_Huygens|Christiaan Huygens]] corrected this in work on impact composed in the 1650s and published posthumously, treating collisions in a moving frame and finding both that the vector sum `m·v` is conserved and that for hard bodies the sum of `m·v^2` — later [[Vis_viva|vis viva]] — is conserved too.[^huygens] John Wallis submitted the signed, directional rule to the Royal Society in 1668, when it posed the collision problem publicly.[^wallis1668] ### Momentum [[Isaac_Newton|Isaac Newton]] made the quantity foundational. Definition II of the *Principia* of 1687 defines the quantity of motion as "the measure of the same, arising from the velocity and quantity of matter conjointly", and the second law is a proportionality between impressed force and the change of that quantity, not between force and acceleration.[^newton1687] The modern reading — that conservation follows from the homogeneity of space — arrived only with Noether in 1918, by which time momentum had been generalized twice, into fields and into [[Quantum_mechanics|quantum mechanics]], without losing the property that made it useful.[^noether1918] ## See also - [[Elastic_collision]] - [[Inelastic_collision]] - [[Coefficient_of_restitution]] - [[Newton's_cradle]] - [[Angular_momentum]] - [[Kinetic_energy]] - [[Noether's_theorem]] - [[Four-momentum]] ## References [^richards-p]: Richards, Donald (2001). *Basic Engineering Science: A Systems, Accounting, and Modeling Approach*. Chapter on linear momentum, p. 145 (`P = m·V`). Portal Book 020, https://open.umn.edu/opentextbooks/textbooks/basic-engineering-science-a-systems-accounting-and-modeling-approach [^richards-sys]: Richards (2001), *Basic Engineering Science*, pp. 152–154 (`P_sys = sum(m_j·V_j)` and `V_G = P_sys/m_sys`; the three-particle example of 15, 10 and 5 kg at (5, 5), (3, −5) and (2, −2) m/s giving `P_sys` = 115**i** + 15**j** kg·m/s). The resulting `V_G` = (3.83, 0.50) m/s is derived. Portal Book 020. [^richards-frames]: Richards (2001), *Basic Engineering Science*, p. 146 (velocities must be measured in an inertial frame) and pp. 150–151 (relative velocity `V_B = V_A + V_B/A`, with the forklift on a 5 m/s flatcar giving 2 and 8 m/s). Portal Book 020. [^richards-rate]: Richards (2001), *Basic Engineering Science*, pp. 163–166 (the rate form `dP_sys/dt = sum(F_ext) + sum_in(mdot·V) − sum_out(mdot·V)`; conservation stated as a law rather than an assumption; Newton's three laws recovered as special cases). Portal Book 020. [^os-collisions]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 1*. OpenStax. Chapter 9 "Linear Momentum and Collisions", pp. 417–418 and 420 (collisions classified by what happens to kinetic energy — explosion, inelastic, perfectly inelastic, elastic; the elastic-collision equation is quadratic and carries one unphysical "no collision" root). Portal Book 077, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^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). The outgoing −0.278 and +2.22 m/s, `V_G` = 1.111 m/s, the 37.5 mJ and 16.7 mJ energy bars, the 55.6 % perfectly inelastic loss and the `1 − e^2` scaling are derived from the book's inputs. Portal Book 077. [^os-thor]: Sanny and Ling (2016), *University Physics Volume 1*, Chapter 9, Example 9.12, pp. 420–421 (a hammer swung 10 m in 1 s against a 200 kg train that then moves 2 m in 0.75 s). The hammer mass of about 73 kg and the ≈2.7 kJ of kinetic energy lost are derived. 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; momentum is conserved only across the impact instant when friction acts). The truck's 11.0 m/s, the car's 27.6 m/s and the ≈60 % kinetic-energy loss are derived. Portal Book 077. [^cline-lagrangian]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, revised second edition. Chapters on Lagrangian dynamics, pp. 155–198 (page to pin) — generalized coordinates and their conjugate momenta `p_i = dL/d(qdot_i)`, cyclic coordinates as constants of the motion, and the Hamiltonian formulation in phase space. Portal Book 073, https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics [^likharev-qm]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, pp. 31–39 (`i·hbar·dPsi/dt = −(hbar^2/2m)·d2Psi/dx2 + U·Psi`; `<dx^2>·<dp^2> >= hbar^2/4`; a minimum-uncertainty packet with `dk = 1/(2·dx)`; `v_gr = hbar·k0/m` and the phase velocity at half of it; the packet's spreading law). Portal Book 047, https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^up3-relativity]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3*. OpenStax. Chapter 5 "Relativity", pp. 183–240 (page to pin) — the Lorentz factor, relativistic momentum `p = gamma·m·v`, and the energy–momentum relation. Portal Book 079, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^poynting1884]: Poynting, John Henry (1884). "On the Transfer of Energy in the Electromagnetic Field." *Philosophical Transactions of the Royal Society of London* 175: 343–361. [^noether1918]: Noether, Emmy (1918). "Invariante Variationsprobleme." *Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse*: 235–257. [^buridan]: Buridan, Jean (14th century). *Quaestiones super octo libros Physicorum Aristotelis*, Book VIII (the impetus questions). [^huygens]: Huygens, Christiaan (1703, composed in the 1650s). *De Motu Corporum ex Percussione*. Published posthumously in *Opuscula Postuma*, Leiden. [^wallis1668]: Wallis, John (1668). "A Summary Account of the General Laws of Motion." *Philosophical Transactions of the Royal Society of London*, volume 3. [^newton1687]: Newton, Isaac (1687). *Philosophiæ Naturalis Principia Mathematica*. London: Royal Society. Definition II (the quantity of motion "arising from the velocity and quantity of matter conjointly") and Law II. ## Bibliography - Richards, Donald (2001). *Basic Engineering Science: A Systems, Accounting, and Modeling Approach*. Portal Book 020 — the momentum-accounting chapter that supplies the rate form used throughout this page. - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 1*. OpenStax. Portal Book 077 — Chapter 9 for the collision classification and the worked problems behind the microsim's presets. - Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, revised second edition. Portal Book 073 — conjugate momenta, cyclic coordinates and the Hamiltonian picture. - Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Portal Book 047 — the momentum operator and the wave packet. - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3*. OpenStax. Portal Book 079 — relativistic momentum. ## External links - [Basic Engineering Science: A Systems, Accounting, and Modeling Approach](https://open.umn.edu/opentextbooks/textbooks/basic-engineering-science-a-systems-accounting-and-modeling-approach), Donald Richards — Portal Book 020 - [University Physics Volume 1](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1), OpenStax — Portal Book 077 - [Variational Principles in Classical Mechanics](https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics), Douglas Cline — Portal Book 073 - The Wikipedia pair's external links list further resources on the history of the concept <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Momentum.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Momentum* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Momentum.html" data-title="Momentum"></div> *Built from `MICROSIM_GUIDE/specs/sims/Momentum.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/Momentum) : [Wikitube](https://en.wikitube.io/wiki/Momentum) · pinned revision [1363847606](https://en.wikipedia.org/w/index.php?oldid=1363847606) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P15 · sim pending (matter/Momentum).*