Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it.

In differential form, this is Newton's second law; the rate of change of the momentum of a particle is equal to the instantaneous force F acting on it,[1]

Impulse is measured in the derived units of the newton second (1 N⋅s = 1 kg⋅m/s) or dyne second (1 dyne⋅s = 1 g⋅cm/s) Under the assumption of constant mass m, it is equivalent to write

In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant.

This conservation law applies to all interactions, including collisions (both elastic and inelastic) and separations caused by explosive forces.

From the point of view of another frame of reference, moving at a constant speed u relative to the other, the position (represented by a primed coordinate) changes with time as

In a coordinate system with x, y, z axes, velocity has components vx in the x-direction, vy in the y-direction, vz in the z-direction.

[16] The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a rocket ejecting fuel or a star accreting gas.

When considered together, the object and the mass (dm) constitute a closed system in which total momentum is conserved.

That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem.

Examples where conservation of momentum does not apply include curved spacetimes in general relativity[26] or time crystals in condensed matter physics.

[27][28][29][30] In fields such as fluid dynamics and solid mechanics, it is not feasible to follow the motion of individual atoms or molecules.

Instead, the materials must be approximated by a continuum in which, at each point, there is a particle or fluid parcel that is assigned the average of the properties of atoms in a small region nearby.

In particular, it has a density ρ and velocity v that depend on time t and position r. The momentum per unit volume is ρv.

[34] Including the effect of viscosity, the momentum balance equations for the incompressible flow of a Newtonian fluid are

[37] The flux, or transport per unit area, of a momentum component ρvj by a velocity vi is equal to ρvjvj.

[dubious – discuss] In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero.

The momentum density is proportional to the Poynting vector S which gives the directional rate of energy transfer per unit area:[46][47]

The above results are for the microscopic Maxwell equations, applicable to electromagnetic forces in a vacuum (or on a very small scale in media).

The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once.

[49][50] Newtonian physics assumes that absolute time and space exist outside of any observer; this gives rise to Galilean invariance.

In the special theory of relativity, Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light c is invariant.

Philoponus pointed out the absurdity in Aristotle's claim that motion of an object is promoted by the same air that is resisting its passage.

[58] In 1020, Ibn Sīnā (also known by his Latinized name Avicenna) read Philoponus and published his own theory of motion in The Book of Healing.

He agreed that an impetus is imparted to a projectile by the thrower; but unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, he viewed it as a persistent, requiring external forces such as air resistance to dissipate it.

[59][60][61] In the 13th and 14th century, Peter Olivi and Jean Buridan read and refined the work of Philoponus, and possibly that of Ibn Sīnā.

[61] Buridan, who in about 1350 was made rector of the University of Paris, referred to impetus being proportional to the weight times the speed.

[67][68][69] Galileo, in his Two New Sciences (published in 1638), used the Italian word impeto to similarly describe Descartes's quantity of motion.

[76] In 1687, Isaac Newton, in Philosophiæ Naturalis Principia Mathematica, just like Wallis, showed a similar casting around for words to use for the mathematical momentum.

[78] In 1721, John Jennings published Miscellanea, where the momentum in its current mathematical sense is attested, five years before the final edition of Newton's Principia Mathematica.