Momentum

Momentum is a measure of how hard it is to stop a moving object. It is simply the mass of the object multiplied by its velocity: p = m x v. A heavy slow object can have the same momentum as a light fast object. The bigger the momentum, the harder it is to stop. A lorry rolling at 5 km/h has more momentum than a tennis ball flying at 100 km/h. The law of conservation of momentum says that in any collision or explosion, the total momentum of all the objects involved stays the same.

  • Formulap = m x vMass times velocity
  • Unitkg m/sOr newton-seconds
  • DirectionA vectorHas size and direction
  • ConservationTotal momentum unchangedBefore and after a collision
  • Bigger object, biggerLike a lorry vs a bikeAt the same speed
  • Faster object, biggerLike a bullet vs a ballOf the same mass

What is momentum?

Imagine catching a tennis ball thrown gently. Easy. Now imagine catching the same ball thrown by a bowling machine at 150 km/h. Much harder. The fast ball has more momentum (same mass, much more speed) and needs much more force to stop.

Now imagine trying to stop a lorry rolling slowly down a slope. Even at just 5 km/h, the lorry has so much mass that its momentum is enormous; you cannot stop it without serious force.

Momentum captures both these ideas in one number: heavy things and fast things are hard to stop, and big-and-fast things are even harder.

The formula

The formula for momentum is straightforward:

p = m x v

Where p is momentum (in kg m/s), m is mass (in kg), and v is velocity (in m/s). The letter p stands for "petere", a Latin word meaning to push.

Some examples:

  • A 0.05 kg tennis ball at 30 m/s has momentum 1.5 kg m/s.
  • A 0.05 kg tennis ball at 60 m/s has momentum 3.0 kg m/s.
  • A 70 kg person walking at 1.5 m/s has momentum 105 kg m/s.
  • A 1,500 kg car at 30 m/s has momentum 45,000 kg m/s.
  • A 40,000 kg lorry at 30 m/s has momentum 1,200,000 kg m/s.

Direction matters

Momentum is a vector: it has direction as well as size. A car going north at 30 m/s has different momentum from one going south at 30 m/s. When momenta point in different directions, they can partly or fully cancel out.

If two cars of equal mass collide head-on at equal speeds, their total momentum before collision is zero (equal but opposite). After collision they tend to crumple together and stop, total momentum still zero. Total momentum has been conserved.

Fact Momentum and force are deeply related. A force is exactly the rate of change of momentum: F = change in momentum / time. The longer a force acts, the more momentum it changes. This is why a car crash with a longer crumple zone is safer: the force is spread over a longer time, so the same change in momentum (the cars momentum at impact) is achieved with less force on the passengers.

Conservation of momentum

One of the most powerful rules in physics is the law of conservation of momentum: in any isolated system (where no outside forces act), the total momentum of all the objects together stays the same, no matter what they do to each other.

This applies in:

  • Collisions: when two objects crash together, their total momentum before equals their total momentum after.
  • Explosions: a stationary object explodes into pieces; the pieces fly in opposite directions so that their momenta cancel out, keeping the total still zero.
  • Rocket launches: a rocket throws hot gas downwards; the rocket itself moves up at a speed determined so that the total momentum of (rocket + gas) stays constant.
  • Particle physics: in particle accelerator experiments, momentum is conserved with absolute precision, helping scientists identify new particles by tracking the momenta of debris.

Sports and momentum

Almost every sport is partly about managing momentum:

  • Football: heading or kicking the ball changes its momentum.
  • Cricket: a fast bowler delivers a ball with huge momentum that the batsman must redirect with precise force.
  • Tennis: a powerful serve gives the ball forward momentum the returner has to stop and reverse.
  • Rugby: a heavy player at speed has enormous momentum, hard to stop without team tackling.
  • Boxing: a fast fist gives a punch high momentum and damaging power.
  • Diving: rotating divers conserve angular momentum, spinning faster as they tuck up and slower as they spread out.
Did you know? The reason it is so dangerous to stand in front of a moving train, lorry or large lorry has to do with momentum. Even at low speeds, the huge mass gives them so much momentum that braking distances can be hundreds of metres. Once moving, a heavy vehicle takes a long time to stop. That is why level crossings have such strict rules and why pedestrians should never assume a slow-looking lorry can stop in time.

Why rockets work

A rocket in space cannot push against anything outside itself, so how does it accelerate? The answer is conservation of momentum.

Inside the rocket engine, hot gas is forced out the back at very high speed. The gas leaves with significant momentum in the backward direction. Because total momentum has to stay the same, the rocket must have equal momentum in the opposite (forward) direction. Throwing mass out the back gains the rocket forward momentum.

The more mass the rocket throws out, and the faster it throws it, the more it accelerates. Modern rockets carry tonnes of propellant to throw out at speeds of several km/s, building up the huge momentum needed to reach orbit.

Try this Sit on a wheeled office chair holding a few heavy books. Throw the books forwards as hard as you can. The chair (with you on it) rolls backwards. The books got forward momentum; you and the chair got equal and opposite backward momentum. Same principle as a rocket. The heavier the books, or the faster you throw, the more you roll backwards. You have demonstrated conservation of momentum on wheels.
Deeper dive: how Newtons cradle works

The classic desk toy called Newtons cradle is a perfect demonstration of momentum conservation in action. The toy is a row of 5 (or sometimes 7) identical steel balls hanging in a frame, just touching each other.

Pull one ball back and let it swing into the row. Strangely, instead of all 5 balls swinging forward, only the ball at the far end pops out. The middle balls stay still. The end ball swings up, comes back, hits the row, and the original ball pops out the other side. The pattern repeats, slowly losing energy to air resistance and tiny friction.

The secret is conservation of momentum AND conservation of energy. When the first ball hits the row, it transfers its momentum and energy through the row to the ball at the other end. The momentum has to go to a single ball (not be spread between several), because spreading would conserve momentum but lose some of the energy. Conserving both at once requires exactly one ball coming off at the end at the same speed as the one that hit. Beautiful, simple physics.

If you pull TWO balls back and release, TWO balls swing out the other side, at the same speed. Pull THREE, three swing out. Whatever you put in, the same amount comes out the other side. Every Newtons cradle in every office is a tiny daily reminder of the laws of physics, working perfectly for decades on end.

For more, see Newtons three laws of motion and speed, velocity and acceleration.