Linear momentum is conserved whenever the net external force acting on a system is zero. That single condition is the entire rule. If outside forces cancel out or are absent, the total momentum of everything inside the system stays constant, no matter how violently the objects inside it push, pull, collide, or explode apart.
The Core Condition: Zero Net External Force
Momentum is the product of an object’s mass and its velocity. Newton’s second law, in its original form, says that net external force equals the rate of change of momentum. Flip that around: if the net external force on a system is zero, momentum cannot change. It’s mathematically locked in place.
This doesn’t mean no forces can exist at all. It means the external forces have to cancel each other out. Billiard balls on a table have gravity pulling them down, but the table pushes back up with an equal normal force. The net external force in the horizontal plane is essentially zero (ignoring friction for a moment), so the total horizontal momentum of any group of colliding billiard balls is conserved during the collision.
Internal Forces Don’t Change Total Momentum
Forces between objects inside a system are called internal forces, and they always come in equal-and-opposite pairs. This is Newton’s third law at work. When two hockey pucks collide on frictionless ice, puck A pushes puck B to the right while puck B pushes puck A to the left with the same strength. Puck A loses some momentum, puck B gains exactly that much. The total stays the same.
This cancellation is perfect and automatic. It doesn’t matter how complicated the interaction is. Objects can shatter, stick together, bounce, or detonate. Every internal force has a partner that produces an equal and opposite change in momentum, so the internal changes always sum to zero. Only forces from outside the system can alter the system’s total momentum.
What Counts as an “Isolated System”
Physicists call a system with no net external force a closed or isolated system. In reality, perfectly isolated systems don’t exist. There’s no way to shield against gravity, and electromagnetic forces reach infinitely far. But the concept is still enormously useful because many real situations come close enough.
The key insight is that you get to choose what’s inside your system. If friction from the floor is messing up your momentum calculation, you can expand the system to include the floor (and the Earth it’s attached to). Now friction becomes an internal force, and momentum is conserved again. The trade-off is that the Earth’s change in velocity is so tiny it’s unmeasurable, but mathematically, it’s there.
You can also look at just one direction. A ball thrown horizontally off a cliff has gravity acting on it vertically, so vertical momentum isn’t conserved. But if air resistance is negligible, horizontal momentum is conserved because there’s no net external force in that direction. Momentum is a vector, so conservation can apply along one axis even when it doesn’t apply along another.
Collisions: Elastic vs. Inelastic
Collisions are the classic momentum-conservation scenario, and momentum is conserved in every type of collision as long as external forces are negligible during the impact.
In an elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other without generating heat, sound, or deformation. Two steel ball bearings clicking together on a desktop come close to this ideal.
In an inelastic collision, momentum is still conserved, but some kinetic energy converts into other forms like heat or sound. A car crash is inelastic: the vehicles crumple, glass shatters, and the wreckage is warmer afterward. The most extreme version is a perfectly inelastic collision, where the objects stick together and move as one mass. This loses the maximum possible kinetic energy, but momentum is still fully conserved. The combined object simply moves slower than the faster incoming object did.
This distinction matters because students sometimes confuse momentum conservation with energy conservation. They’re separate rules. Momentum conservation holds in all collisions (given negligible external forces). Kinetic energy conservation is an additional, stricter requirement that only elastic collisions satisfy.
Recoil and Explosions
A system that starts at rest has zero total momentum. If it then breaks apart, the pieces must carry momenta that add up to zero. This is why a gun recoils: the bullet flies forward with momentum in one direction, and the gun kicks backward with equal momentum in the opposite direction. Because the gun is much heavier than the bullet, it moves much slower. If the bullet were somehow heavier than the gun, the gun would recoil faster than the bullet travels.
Rockets work the same way. Exhaust gases shoot out the back with enormous momentum, and the rocket gains equal momentum in the forward direction. No external force is needed. The system of rocket-plus-exhaust conserves momentum even in the vacuum of space, where there’s nothing to push against.
Connection to Center of Mass
A system’s total momentum equals its total mass multiplied by the velocity of its center of mass. So when momentum is conserved, the center of mass moves at a constant velocity (or stays still if it started that way). During a fireworks explosion, the bright fragments fly in every direction, but the center of mass of all those fragments follows the same smooth arc it was on before the explosion. The internal forces of the blast can’t change that trajectory.
When Momentum Is Not Conserved
Momentum fails to be conserved whenever a net external force acts on the system for some period of time. The impulse-momentum theorem quantifies this: the change in momentum equals the net external force multiplied by the time it acts. A soccer ball sitting on the grass has zero momentum. Your foot supplies an external force over a brief contact time, delivering an impulse that gives the ball its new momentum.
Friction is the most common spoiler in real-world problems. A hockey puck sliding across rough ice gradually loses momentum because friction is an external force (from the ice surface, which is outside the puck system) that isn’t balanced by anything in the horizontal direction. Gravity is another: a dropped ball gains downward momentum the entire time it falls because gravity is a net external force on the ball.
In both cases, you can rescue momentum conservation by expanding your system. Include the Earth, and the gravitational or frictional forces become internal. The math stays clean, though the Earth’s resulting velocity change is vanishingly small.
Does It Still Work at Very High Speeds?
At speeds approaching the speed of light, the formula for momentum changes. The simple “mass times velocity” gets multiplied by a factor that grows as speed increases, making momentum rise much faster than you’d expect from everyday experience. But the conservation law itself holds perfectly. In fact, at relativistic speeds, momentum conservation and energy conservation become two sides of the same coin: one implies the other. The rule that zero net external force means constant momentum never breaks down.