How this instrument works
Momentum is mass times velocity, p = mv, measured in kilogram-metres per second, numerically identical to the newton-second — which is the giveaway. Momentum is what force accumulates over time. Shove something for one tenth of one second and you hand it an impulse; that impulse lands in its momentum ledger and nowhere else. Inside an isolated pair of bodies every internal push arrives with an equal and opposite counter-push, so those internal impulses cancel exactly and the sum m₁v₁ + m₂v₂ cannot shift, however violent or brief the contact.
Descartes named a conserved quantity of motion in 1644 but built it from speed rather than signed velocity, so his rule collapsed whenever two bodies met head-on. Three papers laid before the Royal Society in 1668-69 repaired it: John Wallis treated inelastic impact, Christopher Wren the elastic case, and Christiaan Huygens supplied the decisive move — velocity carries direction, and collisions watched from passing boats look different yet obey identical bookkeeping. Emmy Noether gave the deep reason in 1918: momentum survives because empty space has no preferred address, and any physics indifferent to where an experiment sits must keep its total momentum fixed.
Three assumptions sit under this sheet. Motion runs along a single line, so velocities are signed numbers rather than vectors with components — choose a positive direction and stay loyal to it. The pair counts as isolated for the duration of contact, fair for an impact lasting milliseconds and poor for a trolley grinding against friction for a full second. And speeds stay far below light, where p = mv yields to p = γmv; at one tenth of light speed the classical figure already runs half of one percent low.
- Fix a positive direction along the line of motion. Every entry below depends on that choice.
- Enter Mass 1 and Mass 2 in grams, kilograms or tonnes — units convert on the fly.
- Give Mass 1 initial velocity and Mass 2 initial velocity, writing any backward motion as a negative figure.
- Add Mass 1 final velocity, either measured with a photogate or set by whatever collision you are modelling.
- Read Mass 2 final velocity. The working block prints each momentum sum so you can audit both sides.
Worked example — two carts on a laboratory track
A dynamics cart loaded to Mass 1 = 2 kg rolls at Mass 1 initial velocity = 5 m/s into a stationary partner, so Mass 2 = 4 kg and Mass 2 initial velocity = 0. A photogate beyond the impact point clocks the light cart still creeping forward at Mass 1 final velocity = 1 m/s. Momentum beforehand: 2 × 5 + 4 × 0 = 10 kg·m/s. The light cart retains 2 × 1 = 2 of that, leaving 8 kg·m/s for its heavier partner, so Mass 2 final velocity = 8 ⁄ 4 = 2 m/s.
Now audit the energy, because that is where this scenario earns its keep. Kinetic energy before: ½ × 2 × 5² = 25 J. Afterwards: ½ × 2 × 1² + ½ × 4 × 2² = 1 + 8 = 9 J. Sixteen joules — nearly two-thirds — went into crushing a foam bumper, warming it and making noise. Momentum never flinched. That asymmetry between two quantities you might assume behave alike is precisely why this calculation deserves doing on its own.
Questions
Is momentum conserved even when a collision turns messy?
Yes. Momentum survives anything an isolated pair can do to each other — bouncing, sticking, shattering, exploding — because internal forces always arrive in equal and opposite pairs. Kinetic energy is far pickier and holds only in a perfectly elastic impact. If someone says energy vanished in a crash they are right; if they claim momentum did, hunt for the external impulse nobody counted, usually friction, a kerb or a wall.
What unit does momentum use?
Kilogram-metres per second, kg·m/s, numerically identical to the newton-second. SI grants it no special name. For scale: a thrown baseball carries roughly 5.8 kg·m/s, a sprinting adult about 700, a family car on a motorway near 45,000, and a loaded container ship at harbour speed several hundred million. Particle physicists switch to GeV/c, dividing by light speed to keep their figures civilised.
Why does a negative velocity matter so much here?
Because momentum is a vector, and along one line its sign is the whole of its direction. Enter a rebound as positive and the arithmetic quietly mislays twice the momentum involved. A ball striking a wall at 6 m/s and returning at 6 m/s undergoes a velocity change of 12 m/s, not zero — that factor of two is the single most common slip anyone makes with this quantity. Pick a positive direction first, then type.
What if both bodies stick together after impact?
Then they share one velocity, so whatever you enter for Mass 1 final velocity also applies to Mass 2 final velocity. Solving directly gives v = (m₁v₁ᵢ + m₂v₂ᵢ) ⁄ (m₁ + m₂). This perfectly inelastic case sheds more kinetic energy than any other outcome momentum permits, which is exactly why crumple zones are engineered to deform rather than rebound.
Can I use this for a glancing or angled collision?
Not as written. Off-axis impacts require momentum balanced separately along two perpendicular axes, turning one problem into a pair of equations built from velocity components instead of speeds. Keep this sheet for head-on work: air tracks, rail wagons, bumper-to-bumper rear-end reconstructions, and billiards played dead straight.
How do rockets fit into all this?
A rocket is conservation of momentum with mass on the move. Exhaust hurled backwards carries negative momentum, so the vehicle picks up an equal positive amount, with no ground, air or anything else needed to push against. The fixed-mass equation on this page cannot follow that; Tsiolkovsky's rocket equation can, and it descends from precisely the same conservation rule.