Momentum
Momentum is the product of an object's mass and its velocity, and a single Latin verb sits at its root: pellere, meaning to push or drive. A one kilogram model airplane flying due north at one meter per second carries one kilogram meter per second of momentum, measured against the ground. Change the observer and that number changes too. The same aircraft, mass one thousand kilograms, moving at fifty meters per second, has a momentum of fifty thousand kilogram meters per second. Fly it into a five meter per second headwind and that figure drops to forty-five thousand. Both answers are correct. How can a quantity shift with the point of view and still be one of the most stubbornly conserved ideas in all of physics? Why did it take nearly twelve centuries of argument, from a Byzantine philosopher to Isaac Newton, to pin down what a thrown ball actually keeps? And how does a property of billiard balls reach all the way to photons that have no mass at all? Those are the questions ahead.
Direction is what separates momentum from a mere number. Because it has both magnitude and a direction, momentum can predict where objects head and how fast they move after they collide. The momentum of a system of many particles is simply the vector sum of each particle's momentum. Add a third or a fourth particle and the rule extends without complaint. Every such system has a center of mass, a point fixed by the weighted sum of all the positions. If anything in the system is moving, that center of mass generally moves as well, unless the whole arrangement is in pure rotation around it. Take the total mass of the particles, multiply it by the velocity of that center of mass, and you recover the momentum of the entire system. This result carries a name: Euler's first law. The same vector logic governs three dimensions, where velocity splits into components along the x, y, and z axes, and each vector equation quietly stands in for three separate scalar equations.
A closed system, one that swaps no matter with its surroundings and feels no outside force, keeps its total momentum forever fixed. This is the law of conservation of momentum, and Newton's own laws imply it. The reasoning runs through the third law: when two particles interact, the forces between them are equal in magnitude and opposite in direction. Number the particles one and two, and those opposing forces cancel, so whatever momentum one gains, the other loses in exact measure. The total change comes to zero no matter how complicated the force between them is. Pile on more particles and the momentum traded between every pair still sums to nothing. The law holds across elastic and inelastic collisions alike, and even across separations driven by explosive force. It also reaches beyond the situations Newton studied, surviving into relativistic physics and into electrodynamics. Conservation of momentum is the mathematical shadow of a deeper fact: space is homogeneous, the same from one position to the next, a result that is a special case of Noether's theorem.
Pool balls offer one of the cleanest examples of an almost perfectly elastic collision, thanks to their high rigidity. An elastic collision is one where no kinetic energy turns into heat or any other form. Perfect elasticity can happen without contact at all, as in atomic or nuclear scattering, where electric repulsion holds the objects apart. A satellite's slingshot maneuver around a planet counts as one too. The arithmetic carries a tidy surprise. Take two bodies of equal mass, one still and one approaching at speed v. After the collision the moving body stops dead and the other departs at speed v: the two have simply swapped velocities. A stationary elastic sphere struck by a moving one will send both off at right angles. Inelastic collisions tell the opposite story. Here some kinetic energy becomes heat or sound, visible as the crumpled metal of a traffic collision. A bug striking a windshield is perfectly inelastic, the two ending in one shared motion. The coefficient of restitution measures the loss, defined as the ratio of the relative velocity of separation to the relative velocity of approach.
Three newtons due north is what it takes to push a one kilogram model airplane from rest to six meters per second in two seconds. The change in momentum there is six kilogram meters per second, and the rate of that change comes to three newtons. That equality is no accident. Newton's second law, in its differential form, says the rate of change of a particle's momentum equals the instantaneous force acting on it. When a force varies over time, the accumulated change in momentum between two instants is called the impulse, measured in newton seconds, where one newton second equals one kilogram meter per second. Variable-mass objects bend these rules. A rocket ejecting fuel or a star pulling in gas has a mass that is a function of time, and the naive product rule gives the wrong answer for its motion. The correct equation tracks both the object and the ejected or accreted mass, which together form a closed system whose total momentum is conserved. A rocket flies on exactly this principle: it throws propellant outward, and an equal and opposite momentum drives the rocket forward.
Light carries momentum even though photons have no mass, and the solar sail turns that fact into propulsion. In quantum mechanics, momentum becomes a self-adjoint operator acting on the wave function, and the Heisenberg uncertainty principle limits how precisely momentum and position can be known together, since the two are conjugate variables. Special relativity reshapes the picture further. Einstein kept the rule that equations of motion ignore the reference frame but added that the speed of light c is invariant, which contradicts the older Galilean prediction that light's speed could vary between frames. Coordinates in two frames then connect through the Lorentz transformation rather than the Galilean one, scaled by the Lorentz factor gamma. To stay valid, momentum gains a modified form, and at low velocity it closes in on the familiar Newtonian expression. The four-momentum unites energy and momentum, and through Einstein's mass-energy equivalence its conservation guarantees the conservation of both mass and energy at once. In a game of relativistic billiards, a stationary particle struck by a moving one sends the pair off at an acute angle, not the right angle of the everyday world.
Around 530 AD, John Philoponus took aim at Aristotle in a commentary on the Physics. Aristotle had insisted that a thrown ball stays in motion only because the surrounding air keeps pushing it. Philoponus called this absurd, asking how the same air could both drive an object and resist it, and proposed instead that an impetus was handed to the object in the act of throwing. In 1020, Ibn Sina, known in Latin as Avicenna, read Philoponus and offered his own theory in The Book of Healing, arguing that impetus persists and needs an outside force like air resistance to wear it away. Jean Buridan, made rector of the University of Paris around 1350, held that impetus was proportional to weight times speed. The seventeenth century brought numbers and names. Rene Descartes, in his Principles of Philosophy of 1644, defined quantity of motion as size times speed and called the universe's total of it conserved, though he had no concept of mass apart from weight. Christiaan Huygens spotted the flaw, worked out the true laws of elastic collision in a manuscript of 1652 to 1656, and announced them to the Royal Society in 1668. John Wallis stated the conservation law in 1670, Gottfried Leibniz challenged Descartes in 1686, and in 1687 Isaac Newton fixed the modern meaning in his Principia, defining quantity of motion as arising from velocity and quantity of matter conjointly. By 1721 John Jennings recorded momentum in its current mathematical sense, and in 1728 the Cyclopedia spelled it out plainly: the quantity of motion of any body is its velocity multiplied into its mass.
Common questions
What is momentum in physics?
Momentum in Newtonian mechanics is the product of an object's mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction, and its SI unit is the kilogram meter per second, dimensionally equivalent to the newton second.
What is the law of conservation of momentum?
The law of conservation of momentum states that in a closed system, one that exchanges no matter with its surroundings and feels no external forces, the total momentum stays constant. It follows from Newton's laws and applies to elastic collisions, inelastic collisions, and explosive separations alike.
What is the difference between an elastic and an inelastic collision?
In an elastic collision no kinetic energy is converted into heat or other forms, as with two highly rigid pool balls. In an inelastic collision some kinetic energy becomes heat or sound, as seen in the damage from a traffic collision, and a bug hitting a windshield is a perfectly inelastic case where both bodies end with the same motion.
Who first developed the concept of momentum?
John Philoponus introduced an early version around 530 AD, proposing that an impetus is imparted to a thrown object, in contrast to Aristotle's claim that air keeps it moving. The idea was refined by Ibn Sina in 1020 and Jean Buridan around 1350, before Isaac Newton fixed its modern meaning in his Principia in 1687.
Why does momentum depend on the frame of reference?
Momentum is a measurable quantity whose value depends on the observer's frame of reference. An aircraft of mass one thousand kilograms moving at fifty meters per second has a momentum of fifty thousand kilogram meters per second, but in a five meter per second headwind its momentum relative to the Earth drops to forty-five thousand, and both calculations are equally correct.
How does momentum work for light and photons?
Photons carry momentum even though they have no mass, which makes applications such as the solar sail possible. In relativity the energy-momentum relationship holds for massless particles, and in quantum mechanics momentum becomes a self-adjoint operator on the wave function, bound to position by the Heisenberg uncertainty principle.