What is momentum? Mass in motion, and why it's conserved

Momentum is an object's mass multiplied by its velocity: p = m × v. It measures how much motion something carries and in which direction, and so how hard it is to stop. A truck rolling at walking pace carries far more momentum than a fast baseball. Its most useful property is that it is conserved: in any collision or explosion with no outside push, the total momentum before equals the total after.

In short

• Momentum is mass times velocity, measured in kilogram meters per second. It has a direction as well as a size.
• The total momentum of a group of objects stays the same unless an outside force acts, however they crash, stick or fly apart.
• Changing momentum takes force multiplied by time, called impulse. Stretching out the time, as an airbag does, shrinks the force.

What does momentum measure?

Momentum is the quantity of motion. Newton called it exactly that, and his second law, as physicists now write it, says that force is the rate at which momentum changes. A 10,000 kg truck rolling at 1 m/s has a momentum of 10,000 kg·m/s. A 0.145 kg baseball pitched at 40 m/s has 5.8 kg·m/s; to match the truck it would have to fly at about 69,000 m/s. That is why a slow truck is still hard to stop.

Momentum is written p = m × v, where p is momentum in kilogram meters per second (kg·m/s), m is mass in kilograms and v is velocity in meters per second. Velocity is speed with a direction, so momentum has a direction too: it is a vector. A ball moving east at 5 m/s and an identical ball moving west at 5 m/s have equal and opposite momentum, and together their total is zero.

Momentum is not the same as inertia. Inertia is an object's resistance to any change in its motion, and it depends on mass alone; a parked truck has plenty of inertia and no momentum. Momentum depends on how fast and in which direction the mass is moving.

Why is momentum conserved?

Momentum is conserved because forces come in pairs. By Newton's third law, when two objects push on each other, the pushes are equal and opposite and last exactly as long. So whatever momentum one object gains, the other loses. Inside a closed system, momentum only ever moves from one object to another; the total never changes. Only a force from outside the system can change it.

In modern physics the reason runs deeper: conservation of momentum follows from the fact that the laws of physics are the same everywhere in space. No experiment has ever found it broken.

A worked example: two carts collide

A 2 kg cart rolls east at 3 m/s toward a 1 kg cart standing still on a smooth track. Before the collision the total momentum is 2 × 3 + 1 × 0 = 6 kg·m/s, east.

If the carts stick together, they move off as one 3 kg object. Momentum must still be 6 kg·m/s, so their speed is 6 ÷ 3 = 2 m/s. Now check the energy of motion, kinetic energy, which is ½ × m × v². Before: ½ × 2 × 3² = 9 joules. After: ½ × 3 × 2² = 6 joules. Three joules have gone into heat, sound and bending the latch. Momentum was conserved; kinetic energy was not.

If instead the carts bounce off each other perfectly, a collision physicists call elastic, both totals survive. The arithmetic gives the 2 kg cart 1 m/s and the 1 kg cart 4 m/s, both still heading east. Check: 2 × 1 + 1 × 4 = 6 kg·m/s, and ½ × 2 × 1² + ½ × 1 × 4² = 1 + 8 = 9 J. Real collisions fall between these two cases, and momentum is the one total that holds in all of them.

How does a rocket use momentum?

A rocket works by throwing mass backward. Before the engine fires, rocket and fuel together have zero momentum. Burning fuel shoots hot gas out of the nozzle at several kilometers a second. The gas now carries backward momentum, so to keep the total at zero the rocket must gain exactly as much forward momentum.

A skater shows the same thing on a small scale. A 60 kg skater standing on ice throws a 0.5 kg ball forward at 20 m/s, giving it 10 kg·m/s. The skater slides backward with the same 10 kg·m/s: 10 ÷ 60, about 0.17 m/s. Nothing needs to be pushed against, which is why a rocket works in the vacuum of space, where there is no air or ground.

How is momentum different from kinetic energy?

Both grow with mass and speed, and they are easy to confuse. They answer different questions: momentum says how hard something is to stop over time; kinetic energy says how much work it can do, or damage it can cause, over a distance.

	Momentum	Kinetic energy
Formula	p = m × v	KE = ½ × m × v²
Unit	kg·m/s	joule (J)
Direction	Yes, a vector	No, a single number
Double the speed	Twice as much	Four times as much
In a collision	Always conserved, if no outside force	Conserved only if the collision is elastic; otherwise some becomes heat and sound
Changed by	Force × time (impulse)	Force × distance (work)

The squared speed is why a car at 100 km/h needs about four times the braking distance of one at 50 km/h, not twice: the brakes must remove four times the energy, working over distance.

Why do airbags and bent knees reduce the force?

The change in momentum equals force multiplied by the time it acts: Δp = F × Δt, a quantity called impulse. To stop a moving body you must remove all its momentum, and that amount is fixed. The longer the stop takes, the smaller the force.

Take a 70 kg person in a car at 50 km/h, about 14 m/s, who stops in a crash. Their momentum is 70 × 14 = 980 kg·m/s. Hitting a dashboard, they might stop in about 0.01 seconds: F = 980 ÷ 0.01 = 98,000 newtons. Held by a seatbelt and airbag that stretch the stop to about 0.1 seconds, the force is 980 ÷ 0.1 = 9,800 N, ten times smaller. These figures are rounded for illustration.

The same rule explains why you bend your knees when you land from a jump, why boxers roll with a punch, why egg cartons are padded and why gymnastics mats are thick. Each one makes the stop last longer.

Where does the simple formula stop working?

p = m × v is exact enough for anything slower than a small fraction of the speed of light. Near light speed, momentum grows faster than the formula says and has no upper limit, while speed can never reach light's. That is one way to see why no amount of push gets a spacecraft to light speed.

Light itself has no mass but does carry momentum: each photon's momentum is its energy divided by the speed of light. It is tiny, but sunlight pressing on a large, thin mirror can steer a spacecraft; such solar sails have been tested in space.

Spinning objects carry a related quantity, angular momentum, which is conserved in the same way. A figure skater spins faster by pulling in her arms, and a collapsing cloud of gas spins faster as it shrinks, which is why the planets of the solar system all orbit the Sun in the same direction, inherited from the disk they formed in.

Outside physics, "momentum" is borrowed for sports teams on a winning run and for markets in which rising prices tend to keep rising. Those uses keep the image of motion that is hard to stop, and nothing else of the physics.

Check yourself

• Without looking back: why does the total momentum of two colliding carts stay the same, even when they crumple and stick together?
• What happens to a rocket's momentum when it throws gas backward, and why doesn't it need anything to push against?
• Explain how an airbag reduces the force on a passenger without changing how much momentum they lose.

Sources

• 9.1 Linear Momentum, in University Physics Volume 1 — OpenStax, Rice University
• 9.2 Impulse and Collisions, in University Physics Volume 1 — OpenStax, Rice University
• 9.4 Types of Collisions, in University Physics Volume 1 — OpenStax, Rice University
• 9.7 Rocket Propulsion, in University Physics Volume 1 — OpenStax, Rice University
• Momentum — Wikipedia