A 2 kg trolley rolling at 3 m/s hits a stationary 5 kg trolley and they end up moving together at 0.857 m/s — a fifth of the original speed, for a body now seven times as massive. Nothing about the collision was engineered; that slowdown is the conservation of momentum operating by itself, and it is the reason heavy things are hard to move.
The definition, and why it is a vector
p = m × v, in units of kg·m/s.
Momentum is a vector, so direction is part of the number. A 1 kg object at +5 m/s and a 1 kg object at −5 m/s have total momentum zero, not ten — they are equal and opposite, and that is a physically different situation from two objects both moving right. Sign conventions are not optional bookkeeping here; the momentum calculator takes signed velocities for exactly this reason.
Conservation and its one condition
Total momentum before = total momentum after, provided no significant external force acts during the collision.
The reason is Newton's third law. During contact, object A pushes B one way and B pushes A equally the other way, so the internal forces cancel and nothing external is left to change the total. "No significant external force" is doing real work in that sentence, though — a bouncing ball pushing off the ground exchanges momentum with the ground, which is external to the ball, and the ball's own momentum changes.
Worked example: the trolleys
Before: p₁ = 2 × 3 = 6, p₂ = 5 × 0 = 0, total 6 kg·m/s.
After sticking together: (2 + 5)v = 6, so v = 6/7 ≈ 0.857 m/s — in the original direction of the first trolley.
The total momentum is identical; the combined object simply has more mass to carry it, so it must move more slowly. The everyday versions are the same: a bullet fired into a block of clay leaves the pair moving at a crawl, and a moving car hitting a stationary one is pushed only slightly slower. Meanwhile the energy story is very different, which is the next section.
Elastic versus inelastic
Momentum is conserved in every collision. Kinetic energy is not, and how much survives is what we name the collision:
- Perfectly elastic — both momentum and kinetic energy conserved. Near-idealised: a good billiards break, superelastic bounces. Momentum transfers through the objects, speeds are exchanged rather than averaged.
- Perfectly inelastic — momentum conserved, maximum kinetic energy lost. The trolleys, and anything that sticks. The lost energy is not gone; it became heat, sound and internal deformation.
- Partially inelastic — everything real. A car crash sits here, deforming and heating, and that is precisely what makes crumple zones protective.
For a partially elastic collision the fraction of kinetic energy retained is the coefficient of restitution, e = speed of separation ÷ speed of approach, ranging from 1 (elastic) down to 0 (fully inelastic). It is measured in a lab, and it is a real material property of the collision, not a figure of speech.
The "same total momentum" does not mean the same outcome
Two cars colliding at an intersection can have the same total momentum in wildly different ways. A glancing blow exchanges little and both cars continue roughly on their original paths. A head-on collision with comparable masses brings both to nearly a stop. Same conserved quantity, completely different consequences — because momentum is conserved while kinetic energy, which decides how violent the stop is, is not.
This is the practical point. Insurance premiums, crash test ratings and road-safety design are all about managing the energy, while momentum conservation is the bookkeeping constraint you cannot violate.
Impulse: the same law, applied to safety
The change in momentum equals the average force times the contact time:
Δp = F × Δt
For a fixed required change in momentum, a longer stopping time means a smaller force. This is the entire reason for crumple zones, airbags, helmets and padding on guardrails. A car stopping from 20 m/s in 0.5 s needs F = mΔv/Δt; extending that to 2.5 s with a crumple zone cuts the average force by a factor of five, even though the same total energy has to go somewhere. It goes into deforming the metal instead of the occupants, which is precisely the design goal.
Sign conventions in a two-object collision
Set rightward positive. Then an object moving left has negative momentum, and conservation is simply:
m₁v₁ + m₂v₂ = (m₁ + m₂)v_final
where the left side uses initial velocities and the right uses the common final velocity for the sticking case. Equal and opposite is not a special coincidence in this algebra; it is what "total zero" means.
Two-dimensional collisions
Real collisions are rarely along a line, and then the algebra changes shape. Instead of one equation along one axis, there are two — one for x, one for y — and the vector nature of momentum becomes the whole point. For a smooth, elastic collision you conserve momentum in each direction separately and the kinetic energy in total, which gives four equations for four unknowns (two final velocities) and is entirely solvable.
The billiards break is the standard illustration: the incoming cue ball keeps its x-momentum and hands the rest to the object ball, which departs along the line of centres while the cue ball is deflected at an angle determined by where it struck. The apparent "magic" of a good break is momentum conservation in two dimensions, and the reason the object ball goes where the geometry says it must.
Rockets: momentum with nothing to push against
A swimmer pushes water; a car pushes road; a rocket pushes exhaust and has nothing else. The third law works exactly the same way — exhaust leaves the nozzle at high speed downward, and the rocket receives equal momentum upward. The kinetic energy spent pushing the exhaust comes from the fuel's chemical energy, which is why a rocket's mass falls as it climbs and why it accelerates most at the end of a burn when it is lightest.
There is no subtlety here about which force is "really" acting. A rocket in deep space is not being pushed by anything it touches; the exhaust it leaves behind is a real, measurable stream of matter moving away at kilometres per second, and the momentum it carries is exactly what the vehicle gains.
Frequently asked questions
What is momentum and how do I calculate it?
Momentum p = m × v, mass times velocity, in kg·m/s. A 2 kg object at 3 m/s has a momentum of 6 kg·m/s in the direction of motion.
Why is momentum conserved in a collision?
If no net external force acts during the collision, the total momentum before equals the total after. This follows from Newton's third law: every action has an equal and opposite reaction, so the internal forces cancel.
Is momentum always conserved?
For the total system, yes, provided no net external force acts. It can change if something outside pushes during the collision — a bouncing ball pushing off the ground exchanges momentum with the ground, which is external to the ball.
What is the difference between momentum and kinetic energy?
Momentum is mv and is a vector; kinetic energy is ½mv² and is a scalar. In a perfectly inelastic collision momentum is conserved but kinetic energy is not — the difference becomes heat and sound.