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    Momentum and conservation of momentum

    By Engr. Mehmood Kakar · Updated

    Quick answer

    Momentum = mass × velocity (p = m v), measured in kg m/s. In a closed system, the total momentum before an event equals the total momentum after it. Momentum has a direction, so give one direction a minus sign. Force equals the rate of change of momentum, which is why longer stopping times mean smaller forces.

    What momentum is

    Momentum is mass in motion: p = m × v. A lorry and a bike at the same speed have very different momenta because their masses differ. Momentum is a vector, so it has a direction as well as a size.

    A car of mass 1200 kg travels at 15 m/s. Calculate its momentum.

    1. p = m × v = 1200 × 15
    2. p = 18 000 kg m/s

    Conservation of momentum

    In a closed system, where no external forces act, momentum is conserved:

    total momentum before = total momentum after

    Every momentum question follows the same four steps: choose a positive direction, work out the total momentum before, set it equal to the total after, then solve for the unknown.

    Collisions

    A 2.0 kg trolley moving at 3.0 m/s hits a stationary 1.0 kg trolley. They stick together. Find their velocity after the collision.

    1. Momentum before = 2.0 × 3.0 + 1.0 × 0 = 6.0 kg m/s
    2. Momentum after = (2.0 + 1.0) × v = 3.0 v
    3. 3.0 v = 6.0
    4. v = 2.0 m/s in the original direction

    A 3.0 kg ball moving right at 4.0 m/s hits a 2.0 kg ball moving left at 3.0 m/s. They stick together. Find their velocity afterwards.

    1. Take right as positive.
    2. Momentum before = 3.0 × 4.0 + 2.0 × (−3.0) = 12 − 6.0 = 6.0 kg m/s
    3. Momentum after = (3.0 + 2.0) × v = 5.0 v
    4. v = 6.0 ÷ 5.0 = 1.2 m/s to the right

    The momentum simulation shows a sticking collision with the momentum before and after.

    Explosions and recoil

    Before an explosion everything is at rest, so the total momentum is zero. Afterwards the pieces fly apart with equal and opposite momenta, still adding up to zero.

    A 4.0 kg gun fires a 0.010 kg bullet at 400 m/s. Calculate the recoil velocity of the gun.

    1. Momentum before = 0
    2. Bullet momentum after = 0.010 × 400 = 4.0 kg m/s forwards
    3. Gun momentum must be 4.0 kg m/s backwards: 4.0 × v = 4.0
    4. v = 1.0 m/s backwards

    Force and momentum change

    Force equals the rate of change of momentum: F = m Δv ÷ Δt. (AQA: Higher tier, GCSE Physics only.) For the same change in momentum, a longer time means a smaller force. That is how safety features work:

    • Crumple zones fold up slowly, increasing the time the car takes to stop.
    • Airbags and seat belts increase the time the passenger takes to stop.
    • Cycle helmets, gym mats and soft play surfaces squash, increasing the impact time.

    A 70 kg passenger moving at 15 m/s is stopped by a seat belt in 0.50 s. Without it, a dashboard stops them in 0.10 s. Compare the forces.

    1. Change in momentum = 70 × 15 = 1050 kg m/s
    2. With seat belt: F = 1050 ÷ 0.50 = 2100 N
    3. Without: F = 1050 ÷ 0.10 = 10 500 N
    4. The seat belt makes the force 5 times smaller (2100 N instead of 10 500 N)

    Common mistakes

    • Ignoring direction. Objects moving opposite ways need opposite signs.
    • Forgetting the stationary object. It has zero momentum before, but its mass still counts afterwards if things stick together.
    • Grams. Convert to kilograms: a 10 g bullet is 0.010 kg.
    • Assuming kinetic energy is conserved. Momentum is conserved in every collision, but kinetic energy usually is not: some is transferred to heating and sound.

    Practice

    1. What is the momentum of a 60 kg runner moving at 5.0 m/s?

    p = m × v = 60 × 5.0 → p = 300 kg m/s

    2. A 1.5 kg trolley at 2.0 m/s collides with a stationary 0.50 kg trolley and they stick. Find their velocity.

    Before: 1.5 × 2.0 = 3.0 kg m/s → after: 2.0 × v = 3.0 → v = 1.5 m/s

    3. A 50 kg skater at rest pushes off a 70 kg skater, who moves away at 1.0 m/s. How fast does the 50 kg skater move?

    Momentum before = 0 → 70 × 1.0 = 50 × v → v = 1.4 m/s in the opposite direction

    4. A 0.15 kg ball hits a bat at 20 m/s and stops in 0.010 s. What average force does the bat exert?

    F = m Δv ÷ Δt = 0.15 × 20 ÷ 0.010 → F = 300 N

    Frequently asked questions

    What is the equation for momentum?

    Momentum = mass × velocity, p = m v. Mass is in kilograms and velocity in metres per second, so momentum is in kilogram metres per second (kg m/s). It is Higher tier on AQA.

    What does conservation of momentum mean?

    In a closed system (no external forces), the total momentum before a collision or explosion is equal to the total momentum after it.

    Why is momentum a vector?

    Velocity has a direction, so momentum does too. When objects move in opposite directions, take one direction as positive and the other as negative before adding their momenta.

    How do airbags and crumple zones reduce injuries?

    They increase the time taken for the person’s momentum to change to zero. Force equals rate of change of momentum, so the same change spread over a longer time means a smaller force on the body.

    Engr. Mehmood Kakar

    Written by Engr. Mehmood Kakar

    Physics educator and engineer (MSc Physics, B.Ed, M.Ed) with 8 years of hands-on teaching, from college students to teacher training. About the author · LinkedIn (opens in a new tab)

    Checked against the AQA GCSE Physics (8463) specification on 15 November 2026.