Conservation of Momentum
Conservation of momentum is one of the deepest and most useful ideas in physics. It says that in any closed system, free from outside pushes and pulls, the total momentum never changes. From the recoil of a rifle to the launch of a spacecraft, this single principle lets you predict the outcome of collisions and explosions without knowing the messy details of the forces involved.
What momentum actually is
Momentum is the product of an object’s mass and its velocity. A heavy truck rolling slowly and a light bullet moving fast can carry similar momentum. Crucially, momentum is a vector: it has direction as well as size, so a ball moving north and an identical ball moving south have equal and opposite momenta.
The total momentum of several objects is just the vector sum of each individual p. To add momenta you must respect direction, which is why we often work with components along the x and y axes separately.
Why momentum is conserved
Conservation of momentum follows directly from Newton’s third law. When two objects interact, they push on each other with equal and opposite forces for exactly the same length of time. Each force delivers an “impulse” (force multiplied by time) that changes each object’s momentum, but because the forces are equal and opposite, the two momentum changes cancel. The whole system’s momentum is left untouched.
This means internal forces, no matter how violent, can never change the total momentum of a system. Only external forces can. If you understand Newton’s laws, you already hold the seed of momentum conservation.
Internal forces always come in equal and opposite pairs, so they cancel in the total. The momentum of an isolated system is fixed forever, no matter what chaos happens inside it.
Collisions: elastic and inelastic
Momentum is conserved in every collision where outside forces are negligible, but kinetic energy is not. Physicists sort collisions into two families:
- Elastic: both momentum and kinetic energy are conserved. Idealised billiard balls and gas molecules behave this way.
- Inelastic: momentum is conserved but some kinetic energy is lost to heat, sound, and deformation. A clay ball hitting a wall is the extreme case.
For a perfectly inelastic collision where two objects stick together, conservation gives a tidy result:
Solve for v_f and you know the combined velocity after impact, even though the collision itself was complicated. To explore the energy side of the story, see our kinetic energy calculator.
Recoil, rockets, and explosions
Conservation of momentum shines brightest when something starts at rest and then flies apart. Before a rifle fires, the total momentum is zero. Afterwards the bullet races forward with momentum p, so the rifle must recoil backward with momentum −p to keep the total at zero. The rifle is far heavier, so it moves much more slowly, but the products of mass and velocity match exactly.
Rockets work on the same logic. A rocket hurls exhaust gas backward at high speed; to balance that backward momentum, the rocket gains forward momentum. This is why a rocket accelerates in the vacuum of space with nothing to push against. It pushes against its own expelled fuel.
When momentum is NOT conserved
The catch is the word “isolated.” If an external force acts on your chosen system, total momentum can change. A ball rolling on the ground slows because friction (an external force from the floor) drains its momentum into the Earth. Gravity, friction, and air resistance are common momentum-changers.
The clever fix is to widen your system. Include the Earth, and the momentum the ball loses is gained by the planet. Choose your system so that the forces you care about become internal, and momentum conservation returns as a powerful shortcut.
Frequently asked questions
Is momentum the same as kinetic energy?
No. Momentum is mass times velocity and is a vector; kinetic energy is one-half mass times velocity squared and is a scalar. In a collision momentum is always conserved, but kinetic energy is only conserved if the collision is perfectly elastic.
How can a rocket accelerate with nothing to push on?
It pushes on its own exhaust. The rocket throws mass backward, and conservation of momentum forces the rocket forward. No external surface is needed, which is exactly why rockets work in empty space.
Why does a heavy gun recoil slowly but a light bullet flies fast?
Both carry equal and opposite momentum, but momentum is mass times velocity. The massive gun needs only a small velocity to match the bullet’s momentum, while the tiny bullet needs a large velocity.
