Momentum

The idea in plain language

Momentum is a way of describing an object’s “quantity of motion” that combines its mass and its velocity. The name comes from the Latin movimentum, meaning “movement”, and physicists write it with the letter p.^[1]

The momentum of an object is its mass multiplied by its velocity.^{[1, 10]} Velocity has a direction, so momentum is a vector: it has a size and a direction, and it points the same way the object is moving.^{[6, 10]} In SI units it is measured in kilogram-metres per second (kg·m/s), a combination with no special name of its own.^[10]

OpenStax suggests momentum is perhaps most useful for judging whether an object’s motion is hard or easy to change over a short time.^[1] A supertanker carries such a huge mass that a force takes a long time to change its fairly small velocity.^[1] Gas molecules can move very fast, yet their velocities change almost instantly when they collide, mainly because their masses are so tiny.^[1]

Impulse: how a force changes momentum

How much a force changes an object’s motion depends on the size of the force and also on how long it acts.^[2] A force multiplied by the time it acts is called an impulse, and the impulse–momentum theorem says that an impulse changes an object’s momentum by exactly the amount of the impulse.^{[2, 10]} An impulse is a vector measured in newton-seconds, and the average force over the time interval is enough to work it out.^[2]

Why airbags help. The same change of momentum can come from a large force acting briefly or a smaller force acting for longer.^[2] In a crash, the change of momentum is the same with or without an airbag, but the force on the people inside is vastly different.^[2] According to OpenStax, this is why airbags have been required on all passenger vehicles in the United States since 1991.^[2]

Conservation of momentum

When two objects interact, both of their momenta change, by amounts equal in size but opposite in sign.^[3] The forces that the parts of a system exert on each other (internal forces) always add up to zero, so they cannot change the system’s total momentum.^{[10, 3]} An external force such as gravity or friction, on the other hand, does change the momentum of the system as a whole.^{[3, 10]}

MIT calls a system with zero external force an isolated system and shows that its momentum stays constant; OpenStax uses the name closed system.^{[10, 3]} This is the law of conservation of momentum: the total momentum of a closed system is conserved.^{[3, 10]} Momentum can be neither created nor destroyed; forces, acting as Newton’s laws describe, can only change it.^[13] Because momentum is a vector, it is conserved in all three directions at the same time.^[13] OpenStax counts the law, with the conservation of energy, among the foundations of all physics, and says all experimental evidence supports it, from galactic clusters to quarks.^[3]

Using the law starts with choosing the system: to analyse a car crash, for example, the two cars together make up the system.^[3]

Collisions and explosions

Momentum is conserved in every collision or explosion, but these interactions are not all alike; the difference lies in what happens to the system’s kinetic energy.^[4]

• Elastic collisions: objects bounce off each other, moving apart at the same relative speed at which they approached, and the system’s kinetic energy is the same before and after.^[4]
• Inelastic collisions: the system loses kinetic energy. Any collision in which the objects stick together loses the most kinetic energy possible.^[4]
• Explosions: momentum is still conserved, but kinetic energy increases. If the object starts at rest, the momenta of all the pieces must add up to zero afterwards.^[4]

Particle colliders such as the Large Hadron Collider use head-on collisions: they speed particles up to very high momenta in opposite directions, which maximises the creation of “daughter particles”.^[4]

Rockets: moving by throwing mass away

Rockets move forward by expelling gas backwards at high velocity.^[7] The burned fuel gases have mass and velocity, and therefore momentum, so by conservation of momentum the rocket’s momentum changes by the same amount in the opposite direction.^[5] A bottle rocket is a good example: its mass does not stay constant, so its flight can only be analysed through changes in momentum.^[14] OpenStax shows that even when the force on a rocket is constant, its acceleration is not: it keeps increasing.^[5]

Case study: nudging an asteroid

On 26 September 2022, NASA’s DART (Double Asteroid Redirection Test) spacecraft deliberately smashed into Dimorphos, a 560-foot-wide (170-metre-wide) asteroid, at roughly 14,000 miles (22,530 kilometres) per hour.^{[16, 15]} It was the first time humans had deliberately changed how a celestial object moves.^[15]

Early measurements showed that Dimorphos’s orbit around Didymos had shortened by 32 minutes, from 11 hours 55 minutes to 11 hours 23 minutes, give or take about 2 minutes.^[15] A later study put the change at 33 minutes and 15 seconds.^[16]

The impact blasted an estimated 16 million kilograms (35.3 million pounds) of dust and rock off the asteroid.^[17] The recoil from this debris substantially added to DART’s push, a little like air rushing out of a balloon sends it the other way.^[15] The DART team found that the flying rubble gave Dimorphos a shove several times stronger than the hit from the spacecraft itself.^[17]

Scientists measure this with a momentum enhancement factor, β (beta): the ratio of the momentum actually given to the asteroid to the momentum the impactor brought, along the direction of the ejecta’s net momentum.^[18] A study in Nature found β between 2.2 and 4.9, depending on the mass of Dimorphos, and 3.61 (+0.19/−0.25) if Dimorphos and Didymos have equal densities of 2,400 kg per cubic metre.^[18]

The math (optional)

Momentum. For an object of mass m moving with velocity v, the momentum is:^[1]

p = m × v

In an OpenStax example, a 1,400 kg car moving at 15 m/s has a momentum of 21,000 kg·m/s, while an air molecule of 6 × 10⁻²⁵ kg moving at 500 m/s has 3 × 10⁻²² kg·m/s.^[1]

The second law in momentum form. The second law can be written as: force equals the change in momentum with respect to time, which for an object of constant mass reduces to mass times acceleration.^[13] OpenStax likewise notes that the second law can be written either as a rate of change of momentum or as mass times the rate of change of velocity.^[8]

Impulse. With an average force F acting for a time Δt, the impulse is J = F × Δt, and the impulse–momentum theorem says J = Δp, the change in momentum.^[2] Because an impulse equals a change of momentum, impulse (in newton-seconds) and momentum (in kg·m/s) are measured in equivalent units.^{[2, 10]} Our own example: stopping the car above removes all 21,000 kg·m/s of its momentum, which takes an average force of 21,000 ÷ 1 = 21,000 N over 1 second, or 21,000 ÷ 0.1 = 210,000 N over 0.1 second.^{[1, 2]}

Conservation. For two interacting objects, the changes in momentum are equal and opposite, Δp₁ = −Δp₂, so the total momentum of the pair does not change.^[3] Our own example: if a 2 kg lab cart moving at 3 m/s hits an identical cart at rest and they stick together, the momentum before is 2 × 3 = 6 kg·m/s; afterwards the same 6 kg·m/s is shared by 4 kg, so the carts move off at 6 ÷ 4 = 1.5 m/s.^[3]

The rocket equation. For a rocket in deep space, with no gravity or other outside force, and exhaust leaving at speed u relative to the rocket, the gain in speed as the rocket’s mass falls from m₀ to m is Δv = u ln(m₀/m).^[5] It was originally derived by the Russian rocket scientist Konstantin Tsiolkovsky in 1897.^[5]

Common misconceptions

“Momentum is just another word for kinetic energy.” Both combine mass and velocity, but momentum is a vector, while kinetic energy depends on speed and has no direction.^{[1, 11]} In a closed system, momentum is always conserved; kinetic energy may be too, but very often it is not.^{[4, 12]}

“The team has momentum.” In everyday speech, a sports team scoring points is said to “have momentum”; in physics, a train moving at 10 m/s has more momentum than the same train moving at 2 m/s.^[6]

History

Descartes. In his Principles of Philosophy, Descartes argued that the total quantity of motion in the world stays constant, measuring a body’s quantity of motion as its speed times its size.^{[20, 19]} But his quantity used speed, which has no direction, rather than velocity.^[19]

Huygens, Wallis and Wren. Huygens’s work on collisions of elastic bodies showed the errors in Descartes’s laws of impact, and his memoir on the subject was sent to the Royal Society in 1668.^[22] Answering a question the Royal Society had posed on impact, Huygens proved by experiment that the momentum in a fixed direction before two bodies collide equals the momentum in that direction afterwards; Wallis and Wren also answered the question.^[22]

Newton. According to MacTutor, Newton had early versions of his three laws of motion by 1666 and published the Principia in 1687.^[23] He called momentum the “quantity of motion”, a way of combining an object’s velocity and mass.^{[6, 9]} In an English translation quoted in MIT’s notes, his definition reads: “The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly.”^[9] The principle we now call conservation of linear momentum appears in the Principia as Corollary 3.^[21]

Did Newton write the second law in terms of momentum? Sources differ. OpenStax says Newton actually stated his second law in terms of momentum: the rate at which a body’s momentum changes equals the net force acting on it.^[6] The Stanford Encyclopedia of Philosophy is more cautious. It notes that the modern F = ma form appears in no edition of the Principia, and that Newton’s own wording says a change in motion is proportional to the force impressed.^[21] It also asks what Newton meant by “a change in motion”, since a change in momentum would properly have been called a change in the quantity of motion.^[21]

Leibniz. Leibniz publicly attacked Descartes’s conservation principle and argued that vis viva (“living force”), mv², was an adequate measure of force.^[20]

Symmetry (an aside). Emmy Noether’s theorem proves a relationship between symmetries in physics and conservation principles.^[24] In a teaching paper posted as an arXiv preprint, C. T. Hill and L. M. Lederman state that the conservation law corresponding to space-translation symmetry is the law of conservation of momentum.^[25]

Going further

Momentum builds on Newton’s Laws of Motion. Kinetic energy belongs to Energy & Work, Rockets follows momentum into spaceflight, and Isaac Newton tells his story.

This page covers linear momentum only; angular momentum, and momentum in relativity and quantum physics, are separate topics. Chapter 9 of the free OpenStax University Physics Volume 1 and MIT’s 8.01SC course, listed below, give many more worked examples.