In classical mechanics, momentum (pl. momenta; SI unit kg·m/s, or, equivalently, N·s) is the product of the mass and velocity of an object (p = mv). For more accurate measures of momentum, see the section "modern definitions of momentum" on this page. It is sometimes referred to as line momentum to distinguish it from the related subject of angular momentum. Linear momentum is a vector quantity, since it has a direction as well as a magnitude. Angular momentum is a pseudovector quantity because it gains an additional sign flip under an improper rotation. The total momentum of any group of objects remains the same unless outside forces act on the objects (law of conservation of momentum).
mōmentum was not merely the motion, which was mōtus, but was the power residing in a moving object, captured by today's mathematical definitions. A mōtus, "movement", was a stage in any sort of change, while velocitas, "swiftness", captured only speed. The Romans, due to limitations inherent in the Roman numeral system, were unable to go further with the perception.
The concept of momentum in classical mechanics was originated by a number of great thinkers and experimentalists. The first of these was Ibn Sina (Avicenna) circa 1000, during the Islamic Renaissance who referred to impetus as proportional to weight times velocity. René Descartes later referred to mass times velocity as the fundamental force of motion. Galileo in his Two New Sciences used the Italian word "impeto."
The question has been much debated as to what Isaac Newton's contribution to the concept was. The answer is apparently nothing, except to state more fully and with better mathematics what was already known. The first and second of Newton's Laws of Motion had already been stated by John Wallis in his 1670 work, Mechanica slive De Motu, Tractatus Geometricus: "the initial state of the body, either of rest or of motion, will persist" and "If the force is greater than the resistance, motion will result". Wallis uses momentum and vis for force.
Newton's Philosophiæ Naturalis Principia Mathematica, when it was first published in 1686, showed a similar casting around for words to use for the mathematical momentum. His Definition II defines quantitas motus, "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum. Thus when in Law II he refers to mutatio motus, "change of motion", being proportional to the force impressed, he is generally taken to mean momentum and not motion.
It remained only to assign a standard term to the quantity of motion. The first use of "momentum" in its proper mathematical sense is not clear but by the time of Jenning's Miscellanea in 1721, four years before the final edition of Newton's Principia Mathematica, momentum M or "quantity of motion" was being defined for students as "a rectangle", the product of Q and V where Q is "quantity of material" and V is "velocity", s/t.
Some languages, such as Italian, still lack a single term for momentum, and use a phrase such as the literal translation of "quantity of motion".
If an object is moving in any reference frame, then it has momentum in that frame. It is important to note that momentum is frame dependent. That is, the same object may have a certain momentum in one frame of reference, but a different amount in another frame. For example, a moving object has momentum in a reference frame fixed to a spot on the ground, while at the same time having 0 momentum in a reference frame attached to the object's center of mass.
The amount of momentum that an object has depends on two physical quantities: the mass and the velocity of the moving object in the frame of reference. In physics, the usual symbol for momentum is a small bold p (bold because it is a vector); so this can be written
Example: a model airplane of 1 kg travelling due north at 1 m/s in straight and level flight has a momentum of 1 kg m/s due north measured from the ground. To the dummy pilot in the cockpit it has a velocity and momentum of zero.
According to Newton's second law, the rate of change of the momentum of a particle is proportional to the resultant force acting on the particle and is in the direction of that force. In the case of constant mass, and velocities much less than the speed of light, this definition results in the equation