3.4.1.6MechanicsCore

Momentum

Momentum measures how hard it is to stop something moving — it is the product of an object’s mass and its velocity, $p = m v$ . Because velocity is a vector, so is momentum, and in any closed system the $total$ momentum is conserved. That single idea solves collisions, explosions and recoil problems that would be almost impossible with forces alone.

45 min Video by Science Shorts 3.4.1 Force, energy and momentum

Momentum — GCSE & A-Level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Define linear momentum and state its unit
Express Newton’s second law as the rate of change of momentum
State and apply the principle of conservation of momentum
Define impulse and relate it to the change in momentum
Interpret the area under a force–time graph as impulse
Distinguish between elastic and inelastic collisions

Linear momentum

$Linear momentum$ is defined as mass × velocity. It is a vector in the same direction as the velocity, and its unit is the kilogram metre per second (kg m s⁻¹), equivalently the newton second (N s).

Momentum is the quantity Newton actually used in his second law: the resultant force on an object equals the $rate of change of its momentum$ . For constant mass this reduces to the familiar $F = ma$ .

p = m v

Momentum = mass × velocity (kg m s⁻¹).

F = \frac{Δ p}{Δ t} = \frac{Δ ( m v )}{Δ t}

Resultant force is the rate of change of momentum.

Tip — Always choose a positive direction before you start. A ball moving left at 4 m/s has a momentum of the same size but opposite sign to one moving right at 4 m/s — the sign carries the physics.

Conservation of momentum

The $principle of conservation of momentum$ states that for a closed system — one with no external resultant force — the total momentum before an interaction equals the total momentum after it.

It follows directly from Newton’s third law: during a collision each object exerts an equal and opposite force on the other for the same time, so they receive equal and opposite changes in momentum that cancel out. This works for collisions, for explosions, and for recoil.

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

Total momentum before = total momentum after (u = before, v = after).

1Momentum before

= (2.0 \times 3.0) + (1.0 \times 0) = 6.0

kg m/s.

2After the collision they move together with mass

2.0 + 1.0 = 3.0

kg and velocity

v

3.0 v = 6.0 \Rightarrow v = 2.0

m/s in the original direction.

Answer

v = 2.0

m/s

Impulse

When a resultant force acts for a time, it delivers an $impulse$ equal to the change in momentum it produces. Rearranging Newton’s second law gives impulse = force × time = change in momentum.

This is why follow-through, crumple zones and airbags matter: increasing the time over which a momentum change happens reduces the force. The same $Δ p$ spread over a longer $Δ t$ means a smaller $F$ .

F Δ t = Δ p = m v - m u

Impulse (N s) equals the change in momentum.

1Take the initial direction as positive:

u = + 20

m/s,

v = - 15

m/s.

2Impulse

= Δ p = m (v - u) = 0.15 \times (- 15 - 20)

= 0.15 \times (- 35) = - 5.25

N s, so

5.25

N s directed away from the wall.

Answer

5.25

N s (away from the wall)

Tip — On a force–time graph, the area under the line is the impulse (and therefore the change in momentum). This lets you handle forces that vary during an impact.

Elastic and inelastic collisions

Momentum is conserved in $every$ collision in a closed system, but kinetic energy is not. A collision is $elastic$ if the total kinetic energy is also conserved, and $inelastic$ if some kinetic energy is transferred to other stores (heat, sound, deformation).

A $perfectly inelastic$ collision is one where the objects stick together and move off as one — this loses the most kinetic energy consistent with conserving momentum. Collisions between everyday objects are inelastic; collisions between smooth hard spheres or gas molecules are close to elastic.

Tip — To test whether a collision is elastic, work out the total ½mv² before and after. If they are equal, it is elastic; if kinetic energy has fallen, it is inelastic — but momentum still balances either way.

Equation recap

p = m v

Linear momentum = mass × velocity.

F = \frac{Δ p}{Δ t}

Force = rate of change of momentum.

F Δ t = Δ p

Impulse = force × time = change in momentum.

E_{k} = \frac{1}{2} m v^{2} = \frac{p ^{2}}{2 m}

Kinetic energy, useful for checking elastic collisions.

Common mistakes to avoid

Treating momentum as a scalar and ignoring direction.

Momentum is a vector. Assign a positive direction and give opposite velocities opposite signs before adding.

Assuming kinetic energy is conserved in every collision.

Momentum is always conserved in a closed system, but kinetic energy is only conserved in an elastic collision.

Confusing impulse with force.

Impulse is force × time (unit N s) and equals the change in momentum — not the force itself.

Applying conservation of momentum when an external force acts.

Momentum is only conserved for a closed system with no external resultant force (e.g. ignore friction over a short collision).

Key takeaways

Momentum p = mv is a vector measured in kg m/s (or N s).
In a closed system total momentum is conserved in collisions, explosions and recoil.
Impulse = FΔt = change in momentum, and equals the area under a force–time graph.
All collisions conserve momentum; only elastic collisions also conserve kinetic energy.

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