Momentum – GCSE Combined Science Physics

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Imagine a football rolling gently towards you, and then a bowling ball rolling at the same speed. You would happily stop the football with your foot. You would not try that with the bowling ball. Something about the bowling ball makes it much harder to stop.

That something is momentum. Momentum is a measure of how hard it is to stop a moving object. It depends on both the mass and the velocity, so a heavy, fast object has a lot and a light, slow one has very little.

Here is a picture to keep in mind. An empty shopping trolley is easy to stop. A full one rolling at the same speed is much harder to stop. The full trolley has more momentum because it has more mass.

Momentum is useful because it is conserved. In a crash or an explosion, momentum is not lost. It is passed from one object to another, and that lets us predict speeds after a collision.

By the end you will be able to do three things. You will calculate momentum and give it a direction. You will use conservation of momentum for collisions and for recoil. You will explain how airbags and crumple zones make a crash safer.

A note on units. Mass is in kilograms, velocity is in metres per second and momentum is in kg m/s. One part near the end, force = change in momentum ÷ time, is Higher tier, and it is labelled when we reach it.

What momentum is

Think of rolling a tennis ball and a bowling ball towards your foot at the same speed. The bowling ball is much harder to stop. The reason is that it has more momentum.

Momentum depends on two things, the mass and the velocity. A heavier object has more momentum than a lighter one moving at the same speed. A faster object has more momentum than a slower one with the same mass.

momentum = mass × velocity, or p = m × v in symbols. The mass is in kg and the velocity is in m/s, so the unit of momentum is kg m/s. Use it whenever you know the mass and the velocity.

A lorry of 4000 kg moving at 10 m/s has a momentum of 4000 × 10 = 40 000 kg m/s. A car of 1000 kg at 20 m/s has 20 000 kg m/s, and a cyclist and bike of 80 kg at 10 m/s has 800 kg m/s. The lorry has by far the most, even though the car is faster.

A heavy object moving slowly can have the same momentum as a light object moving fast. A 300 kg quad bike at 2 m/s has 600 kg m/s. So does a 60 kg sprinter running at 10 m/s.

Momentum has a direction

Velocity has a direction, so momentum has one too. Momentum is a vector. It points the same way as the velocity.

To handle direction in a sum, choose one direction to be positive. Say right is positive. An object moving right then has positive momentum, and an object moving left has negative momentum.

A 2 kg trolley moving right at 3 m/s has p = 2 × 3 = +6 kg m/s. An identical trolley moving left at 3 m/s has p = 2 × (−3) = −6 kg m/s. The size is the same and the sign is different, and the sign tells you which way the trolley is going.

If both trolleys are on the same track, the total momentum is +6 + (−6) = 0. A total of zero does not mean that nothing is moving. It means the two momenta cancel each other out.

Always write the sign or the direction in your answer, because a size without a direction is only half of a vector. When a question gives two objects moving towards each other, the minus sign is where marks are won or lost. Say clearly which way you chose to be positive.

Conservation of momentum

Imagine a game of pool. A moving white ball hits a still red ball, so the white ball slows down and the red ball shoots away. Momentum has moved from one ball to the other.

A closed system is a group of objects with no outside forces acting on them. In a closed system, the total momentum before an event equals the total momentum after it. This is called the conservation of momentum, and it works for collisions and for explosions.

To use it, work out the momentum of each object before and add them up. Then do the same for after. Choose a positive direction first, and give an object going the other way a minus sign.

Here is an example. A 1 kg trolley at 4 m/s hits a 3 kg trolley that is still, and they stick together. Before, the total momentum is 1 × 4 + 0 = 4 kg m/s, and after, the mass is 4 kg, so 4 × v = 4 and v = 1 m/s.

Momentum is conserved, but kinetic energy usually is not. In a crash some of the energy goes to sound and to heat. So never say that kinetic energy stays the same in a collision.

Explosions and recoil

Think of a gun firing. Before the shot, the gun and the bullet are both still. Their total momentum is zero.

When the gun fires, the bullet shoots forwards with momentum. Momentum is conserved, so the total must still be zero. The gun must therefore get the same amount of momentum backwards, and this backwards movement is called recoil.

Take forwards as positive. The bullet has +p and the gun has −p, so the two add up to zero. The sizes are equal and only the signs differ.

The heavier object moves more slowly. A rifle of 4 kg fires a 0.02 kg bullet at 300 m/s. The bullet has 0.02 × 300 = 6 kg m/s, so the rifle has −6 kg m/s and recoils at 6 ÷ 4 = 1.5 m/s backwards.

The same thing happens when you jump off a skateboard, or when a swimmer pushes off the wall of a pool. You go one way and the board goes the other. Two ice skaters who push each other apart start with zero total momentum and finish with equal and opposite momentum.

Force and changing momentum (Higher tier)

Higher tier. A force changes the momentum of an object. The bigger the force, the faster the momentum changes.

force = change in momentum ÷ time taken, or F = Δp ÷ Δt in symbols. The force is in N, the change in momentum is in kg m/s and the time is in s. Use it when you know how much the momentum changes and how long it takes.

This is the same as F = m × a written another way. The change in momentum is the mass × the change in velocity. Divide that by the time and you get the mass × the acceleration.

Here is an example. A football of mass 0.4 kg is kicked from rest and leaves at 20 m/s, so the change in momentum is 0.4 × 20 = 8 kg m/s. If the kick lasts 0.05 s, the force is 8 ÷ 0.05 = 160 N.

Notice that the time is on the bottom of the fraction. Make the time longer and the force gets smaller, or make it shorter and the force gets bigger. That one fact explains most safety features, as the next step shows.

Crumple zones, airbags and seat belts (Higher tier)

Think about catching a cricket ball. If you hold your hand stiff, it stings, but if you pull your hands back as you catch, it feels much softer. In both cases the momentum of the ball changes by the same amount, and what changes is the time.

From F = Δp ÷ Δt, a longer time means a smaller force. Car safety features use this idea. They make a crash last longer, so the force on the people inside is smaller.

A crumple zone is the front or back of a car that is designed to collapse in a controlled way. As it crumples, the car takes longer to stop. Airbags and seat belts also stretch or squash a little and spread the stop over more time.

Look at the bars. A person's momentum changes by 6000 kg m/s in a crash. If the stop lasts 0.2 s, the force is 30 000 N. If safety features stretch it to 0.6 s, the force is only 10 000 N.

The same idea explains why a gymnast bends their knees on landing. Safer does not mean that the change in momentum is smaller. It means the force is smaller because the time is longer.

Recap

momentum = mass × velocity, in kg m/s. It is a vector, so choose a positive direction and use a minus sign for the other way.

In a closed system, the total momentum before an event equals the total momentum after it. This works for collisions and for explosions.

In an explosion that starts from rest, the total momentum is zero. The pieces fly apart with equal and opposite momentum, so the lighter piece moves faster. The recoil of a gun is an example.

Higher tier: force = change in momentum ÷ time taken. A longer time means a smaller force, so crumple zones, airbags and seat belts protect people by making a crash last longer.

Momentum is conserved in a collision, but kinetic energy usually is not, because some of it is transferred to sound and heat.

How this is asked in an exam: a momentum calculation is usually two marks, one for mass × velocity and one for the answer with its unit. Collision questions ask for a speed after objects stick together, so find the total momentum before and divide by the total mass. Explain questions need a chain: the change in momentum is the same, the time is longer, so the force is smaller.