Conservation of Momentum
Momentum measures how hard it is to stop something moving — it depends on both mass and velocity. When objects in a closed system collide or push off each other, momentum can transfer from one object to another, but the total amount never changes. That's why a fast, heavy ball can send a light resting ball flying, while barely slowing down itself.
The formula
p = m · v
- p — momentum (kg·m/s): how much motion an object carries
- m — mass (kg): how much matter the object has
- v — velocity (m/s): how fast the object moves, and in which direction
Worked example
A 3 kg cart moving at 4 meters per second crashes into a stationary 1 kg cart, and they stick together. How fast do they move after the collision?
- m₁ = 3 kg
- v₁ = 4 m/s
- m₂ = 1 kg
- m₁ · v₁ + m₂ · v₂ = (m₁ + m₂) · v
- 3 kg · 4 m/s + 1 kg · 0 m/s = (3 kg + 1 kg) · v
v = 3 m/s
Test yourself
The same 3 kg cart moving at 4 meters per second now crashes into a heavier 3 kg cart at rest, sticking together. What is their speed after the collision?
- 3 m/s
- 6 m/s
- Correct answer: 2 m/s
Right! Momentum before is 3 kg × 4 m/s = 12 kilogram meters per second, shared over a total mass of 6 kg, giving 2 meters per second.
Two identical 2 kg carts roll toward each other, each at 3 meters per second, collide head-on, and stick together. What is their momentum right after the collision?
- Correct answer: Zero
- 12 kilogram meters per second
- 6 kilogram meters per second
Exactly! Their momenta are equal and opposite, 6 kilogram meters per second each way, so they cancel: 6 minus 6 is zero — the carts stop dead.
Where you see this
A parked car gets shoved forward when rear-ended, and Newton's cradle's swinging balls transfer motion from one end to the other almost perfectly — both are momentum handing off from one object to another. A rocket even works by throwing exhaust gas backward so the total momentum of rocket-plus-gas stays unchanged.
Common mistakes
Students often expect a heavier object to "win" a collision by keeping more of its speed — what's actually conserved is the total momentum (mass times velocity) of the whole system, not the speed of either object alone, and a heavy slow object can hand off a lot of momentum to a light fast one. It's also easy to forget that momentum has a direction: two equal-and-opposite momenta cancel to zero, they don't add.
How it connects
Momentum conservation is Newton's second and third laws — equal and opposite forces during a collision — applied over time to a pair of objects instead of one. Energy conservation, next, is a second, parallel accounting tool for the same collisions; together the two let you solve almost any collision without knowing the forces involved at all.