Momentum and Collisions

What makes a moving object hard to stop?

Imagine catching a tennis ball thrown at you — easy. Now imagine catching a bowling ball thrown at the same speed, or the tennis ball fired from a cannon. Both are far harder to stop. What makes a moving object difficult to stop depends on two things: how much mass it has and how fast it is going. Physicists combine these into a single, powerful quantity called momentum.

Defining momentum

Momentum is defined as mass multiplied by velocity:

p = m × v

where p is momentum (in kg·m/s), m is mass (kg), and v is velocity (m/s). Like velocity, momentum is a vector: it has a direction as well as a size. A truck moving north and an identical truck moving south have momenta that are equal in size but opposite in direction.

Worked example. A 0.16 kg cricket ball is bowled at 35 m/s. Its momentum is:

p = mv = 0.16 × 35 = 5.6 kg·m/s

A heavy, slow lorry and a light, fast bullet can have similar momenta — which is exactly why both are dangerous in a collision.

The big idea: conservation of momentum

Here is one of the most important and far-reaching laws in all of physics:

In any closed system — one with no external forces — the total momentum is conserved. The total momentum before an event equals the total momentum after it.

This holds for every collision, every explosion, every push-off, anywhere in the universe. Momentum is never created or destroyed; it only gets transferred between objects.

This law is actually a direct consequence of Newton's laws of motion. When two objects collide, the third law says they push on each other with equal and opposite forces. Equal and opposite forces produce equal and opposite changes in momentum — so whatever momentum one object gains, the other loses, and the total stays fixed.

Worked example — a collision. A 1000 kg car moving at 20 m/s rear-ends a stationary 1000 kg car, and the two lock together. How fast do they move afterward?

Total momentum before:

p = (1000 × 20) + (1000 × 0) = 20,000 kg·m/s

After the crash the combined mass is 2000 kg moving at one speed v. Because momentum is conserved:

20,000 = 2000 × v, so v = 10 m/s

The momentum stayed at 20,000 kg·m/s — it just got shared across twice the mass, halving the speed.

Elastic vs inelastic collisions

All collisions conserve momentum, but they differ in what happens to kinetic energy (the energy of motion, ½mv²).

• Elastic collision: kinetic energy is also conserved. The objects bounce apart cleanly with no energy lost to heat or sound. Snooker balls and gas molecules come very close to this ideal.
• Inelastic collision: kinetic energy is not conserved — some is converted into heat, sound, and the work of deforming the objects. The car crash above is perfectly inelastic because the cars stick together; the "lost" kinetic energy went into crumpling metal, heat, and the bang of the impact.

It is vital to remember: in every collision momentum is conserved, but kinetic energy is only conserved in elastic ones. For more on the different forms energy can take when it transforms, see the forms of energy.

Explosions: momentum from rest

Conservation of momentum also explains explosions and recoil. Before a gun fires, the gun and bullet are at rest, so the total momentum is zero. After firing, the bullet shoots forward with momentum +p, so the gun must recoil backward with momentum −p, keeping the total at zero. Because the gun is much heavier than the bullet, its backward velocity is much smaller — but its momentum is equal and opposite. The same principle launches rockets: hot gas is thrown backward, and the rocket gains an equal and opposite forward momentum.

Impulse: force and time

What actually changes an object's momentum? A force acting over time. This product is called the impulse:

Impulse = Force × time = change in momentum F × t = Δp

This equation has life-saving consequences. To stop a moving body you must remove all its momentum (Δp is fixed by its mass and speed). If you stop it over a longer time, the force needed is smaller.

This is the secret behind car safety:

• Crumple zones make the car deform slowly during a crash, extending the stopping time and cutting the force on the passengers.
• Airbags do the same for your head and chest, spreading the impact over a longer time and a wider area.
• A gymnast bends their knees when landing to stretch out the stopping time and reduce the force on their legs.

In each case the change in momentum is the same, but stretching out the time slashes the peak force. The energy involved here connects to ideas in energy, work, and power.

Try it yourself! 🧪

Experiment 1 — Newton's cradle. A Newton's cradle is five identical metal balls hanging in a row, just touching. Pull one ball back and release it: it strikes the row and one ball swings out the far side at the same speed. Pull back two balls, and two swing out. This is conservation of both momentum and kinetic energy (an almost-elastic collision) in action — the only way to satisfy both laws at once is for the same number of balls to leave at the same speed. If only momentum mattered, other combinations would be possible; the fact that exactly the matching number emerges proves kinetic energy is conserved too.

Experiment 2 — Coin collisions. On a smooth table, flick one coin into a stationary identical coin head-on. The moving coin stops dead and the struck coin shoots off at almost the same speed — momentum transferred completely, just like the snooker break. Now try flicking a small coin into a large one and watch how the heavier coin barely moves: same momentum to share, but spread over more mass means less speed, exactly as the equations predict.