Momentum as quantity of motion

In brief

Two ice skaters stand still, face each other, and push apart. Before the push, neither is moving. After it, both glide backward, the lighter one faster than the heavier one. Nothing here contradicts the fact that each skater, taken alone, clearly gained motion from nothing that skater's own body did in isolation, someone else's push was involved. What this entry builds is the quantity that does not change across the whole event: the combined "quantity of motion" of the pair, defined as mass times velocity and added up with direction taken seriously, was zero before the push and is zero after. That conserved total, not the motion of either skater alone, is momentum, and the reason it is conserved is not a separate law bolted onto Newton's mechanics. It falls straight out of the fact that forces between two bodies always come in equal, opposite, same-duration pairs.

The full treatment

First look: two skaters and a shove

Picture the skaters again, mass m1 and m2, at rest. Skater 1 pushes skater 2 away. During the push, skater 1's hands exert some force on skater 2, and by simple observation, skater 2's hands exert a force back on skater 1, through the exact same contact, for the exact same stretch of time. Afterward skater 1 moves one way at speed v1, and skater 2 moves the other way at speed v2. If skater 2 is heavier, skater 2 ends up moving slower. Something is balancing here beyond a vague sense of fairness, and pinning it down precisely is the job of this entry.

Building the idea: defining momentum

Define the momentum of an object as its mass multiplied by its velocity, written p equals m times v. Velocity is a vector, meaning it carries a direction, so momentum does too: a mass moving left and an equal mass moving right at the same speed have momenta that are equal in size and opposite in direction, and if you insist on adding them as vectors, they cancel to zero. This is not a trick, it reflects something physically true, an observer watching both skaters glide apart sees a system whose center stays put, exactly as if nothing were moving at all when viewed as a whole.

Newton's second law, in its original form from the Principia, states that force equals the rate at which momentum changes, not force equals mass times acceleration directly (the two coincide only when mass stays constant, which is the ordinary case here). Over a short push lasting a time delta-t, a roughly constant force F changes an object's momentum by an amount F times delta-t.

The formal model: deriving conservation from action and reaction

Now apply Newton's third law, that every force comes with an equal and opposite reaction, to the skater push. Let the force skater 1 exerts on skater 2 during the push be F, acting for the duration delta-t of the contact. By the third law, skater 2 exerts a force of exactly minus F, the same size, opposite direction, on skater 1, for that same delta-t, since it is the same contact and the same interval of contact. The change in skater 2's momentum is F times delta-t. The change in skater 1's momentum is minus F times delta-t, the exact negative. Add the two changes together: F times delta-t plus minus F times delta-t equals zero. The total momentum of the two-skater system, m1 v1 plus m2 v2, is therefore unchanged by the push, even though each skater's individual momentum changed enormously, from zero to something large. Conservation of momentum is not an extra assumption here, it is a direct consequence of action and reaction acting for equal times, worked out algebraically rather than declared.

System boundaries: what "isolated" actually means

This derivation only cancels forces that are internal to the system you are tracking, forces the chosen objects exert on each other. If an external force acts during the interaction, friction from the ice, a push from a third person, gravity pulling unevenly, that external force is not paired inside your system and can change the total momentum of your chosen boundary. The fix is not to abandon the law, it is to draw the boundary wider: include the ice and the earth beneath it, and momentum is conserved again, transferred into a body so large that its recoil is unmeasurably small. Choosing the system boundary honestly, deciding what counts as "inside" versus "outside" for a given problem, is the actual skill this entry is asking you to practice.

Lineage

Rene Descartes proposed in 1644 that "quantity of motion," understood loosely as mass times speed with no direction attached, was conserved across all interactions in the universe, a bold but ultimately flawed claim because it ignored direction and so failed for many collisions. Christiaan Huygens, John Wallis, and Christopher Wren independently corrected and refined the picture between 1668 and 1669 in papers presented to the Royal Society, working through the rules for colliding bodies and recognizing that direction had to be built in. Isaac Newton formalized the corrected picture in the Principia in 1687, defining momentum precisely as mass times velocity, stating his second law in terms of its rate of change, and stating the third law of equal and opposite forces that this entry uses to derive conservation directly, rather than treating it as a separate postulate.

The strongest case for it

Momentum conservation explains recoil, explosions, and collisions of every kind, a rifle kicking back when it fires a bullet, a firework scattering fragments whose total momentum still adds to what the shell had before exploding, two vehicles crumpling together in a crash. It underlies rocket propulsion, where a rocket gains forward momentum exactly matching the backward momentum of the exhaust it ejects, with no air to push against required. Beyond classical mechanics, momentum conservation survives essentially unchanged into relativity, with the momentum formula modified for very high speeds, and into quantum mechanics, which is part of why physicists treat momentum, not force, as the more fundamental conserved quantity underneath Newton's laws.

The strongest case against it

The law is only guaranteed for a system with no net external force acting on it; forget to account for an external push, friction with the ground, air resistance, gravity acting only on part of the system, and the momentum of your chosen boundary will appear to change, correctly so, because it genuinely is exchanging momentum with something outside your boundary. A frequent misconception is treating momentum as if it behaved like a simple total amount of "oomph" without direction; two equal masses approaching each other at equal speed carry a combined momentum of zero, not double, because their directions cancel. Another misconception is confusing momentum conservation with kinetic energy conservation; momentum is conserved in every collision inside an isolated system, but kinetic energy is only conserved in special, perfectly elastic cases, a distinction the next entry in this sequence takes up directly.

Where it stands now

Conservation of momentum ranks among the most secure results in physics, unbroken since Newton's formulation and unmodified in form (aside from the relativistic correction to the momentum formula itself) by anything discovered since. It is broad-consensus physics used daily in engineering, astrophysics, and particle collision analysis exactly as derived here.

Test yourself

A loaded cart at rest on a frictionless track suddenly splits into two pieces of known, unequal mass by an internal spring release, and one piece is measured moving off at a known speed. Using conservation of momentum and nothing else, find the speed and direction of the other piece. Then extend the problem: the same cart, instead of starting at rest, is already rolling at a known speed when the split happens. Find both final velocities in that case, and explain, in terms of where you drew your system boundary, why an observer standing on the ground and an observer riding alongside the original cart would each correctly report momentum being conserved, despite disagreeing about the numerical value of the total momentum itself.