What is momentum?

Momentum is a measure of how much motion an object has, and how hard it is to stop. It is mass multiplied by velocity: p = mv, measured in kilogram metres per second (kg m/s). A slow lorry and a fast tennis ball can both have a lot of momentum. Momentum is a vector, so it has a direction, and in any collision or explosion with no outside forces the total momentum stays the same. That rule is called the conservation of momentum.

The momentum formula

The letter p is used because m is already taken by mass. The two units for momentum are the same thing: 1 N s = 1 kg m/s, because a newton is 1 kg m/s².

Example 1: a baseball

A 0.145 kg baseball is thrown at 40 m/s. Its momentum is p = 0.145 × 40 = 5.8 kg m/s.

Example 2: same momentum, different motion

A 2000 kg truck moving at 15 m/s has momentum 2000 × 15 = 30 000 kg m/s. A 1000 kg car at 30 m/s has 1000 × 30 = 30 000 kg m/s too. Half the mass at twice the speed gives the same momentum, so both would take the same impulse to stop.

Is momentum a vector?

Yes. Momentum has the same direction as the velocity, so it is a vector quantity. In a calculation, choose one direction as positive. Anything moving the other way has a negative velocity, and so a negative momentum.

This is what makes conservation of momentum work. Two identical carts moving towards each other at the same speed have a total momentum of zero, because their momenta are equal and opposite. Speed and kinetic energy, by contrast, have no direction, so they never cancel.

Conservation of momentum

The law of conservation of momentum says that in a closed system, where no external forces act, the total momentum before an event equals the total momentum after it.

Here u stands for velocities before and v for velocities after. The rule follows from Newton's third law. During a collision, each object pushes on the other with an equal and opposite force for the same time, so whatever momentum one gains, the other loses.

Example 3: a collision where objects stick together

A 1200 kg car moving at 20 m/s hits a stationary 800 kg car, and the two lock together. How fast do they move just after the crash?

Momentum before: 1200 × 20 + 800 × 0 = 24 000 kg m/s. After, the cars move together with a combined mass of 2000 kg:

They move off at 12 m/s in the original direction.

Example 4: objects moving towards each other

A 2 kg trolley moving right at 3 m/s collides with a 1 kg trolley moving left at 4 m/s. They stick together. Find their velocity afterwards.

Take right as positive, so the second trolley's velocity is −4 m/s.

So v = 0.67 m/s. The answer is positive, so the trolleys move right at 0.67 m/s.

Example 5: an explosion, or pushing apart

Two ice skaters, of 60 kg and 40 kg, stand still together and then push each other apart. The 40 kg skater moves off at 1.5 m/s. How fast does the 60 kg skater move?

The total momentum before is zero, so it is zero afterwards too. The two momenta must be equal and opposite:

The 60 kg skater moves at 1 m/s in the opposite direction. Rockets, recoiling guns and jumping off a boat all work the same way.

Now try a collision yourself, then press Mark to check your answer in the course.

Elastic and inelastic collisions

Momentum is always conserved in a collision, but kinetic energy usually is not.

• In an elastic collision, the total kinetic energy is the same before and after. Collisions between gas molecules are close to elastic.
• In an inelastic collision, some kinetic energy is transferred to other stores: heat, sound, and the energy that bends metal. Objects that stick together are always inelastic.

Check Example 3. The kinetic energy before was ½ × 1200 × 20² = 240 000 J. After, it was ½ × 2000 × 12² = 144 000 J. So 96 000 J went into crumpling the cars, heating them and making noise, while the momentum stayed exactly the same.

Change in momentum and force

A force changes an object's momentum. Newton's second law can be written as force equals rate of change of momentum:

The change in momentum, Δp = mv − mu, is called the impulse, and it also equals FΔt. That is the idea behind crumple zones, airbags and soft landings: spreading the same change in momentum over a longer time makes the force smaller.

Example 6: a ball bouncing off a wall

A 0.06 kg tennis ball hits a wall at 20 m/s and bounces straight back at 15 m/s. It is in contact with the wall for 0.05 s. Find the change in momentum and the average force.

Take the direction towards the wall as positive, so the final velocity is −15 m/s.

The change in momentum is 2.1 kg m/s away from the wall, and the average force is 42 N. Forgetting that the ball reverses direction, and using 20 − 15 = 5 m/s, is the classic mistake here.

Angular momentum

Spinning objects have angular momentum, the rotational version of momentum. For a point mass moving in a circle it is L = mvr, and in general L = Iω, where I is the moment of inertia and ω the angular velocity. It is conserved too: a figure skater who pulls in her arms spins faster, because her moment of inertia falls and her angular velocity rises to keep L the same.

Common mistakes

• Ignoring direction. Momentum is a vector. Give velocities in the opposite direction a minus sign.
• Mass in grams. Convert to kilograms: 145 g is 0.145 kg.
• Thinking kinetic energy is conserved too. Only in elastic collisions. Momentum is conserved in all of them.
• Forgetting to combine masses when objects stick together after a collision.
• Getting the change in momentum wrong for a bounce. A reversal from +20 m/s to −15 m/s is a change of 35 m/s.
• Using conservation when there is an outside force. Friction or a push from outside means the total momentum can change.

Momentum on the formula sheet

AQA's GCSE Physics equations sheet, checked in November 2026, gives momentum = mass × velocity, p = mv, and force = change in momentum ÷ time taken, F = mΔv/Δt. Both are marked as Higher tier, and in AQA's specification momentum is a Higher tier topic, including the idea that momentum is conserved in a closed system. In the US, momentum is a core topic in high school physics and in AP Physics 1, which has a unit on linear momentum.

Check your answers in the Momentum Calculator, which runs collisions for any masses and speeds.

Practise momentum in a short timed test

High School Physics and AP Physics 1 each give you a short timed practice test every day, with 8 questions worth 25 to 35 points. In the UK, the same practice opens as GCSE Physics Higher. The Laws of Motion course has a collision simulator where you can watch momentum being shared.

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