Physics · Further Mechanics · A2 Level

Impulse & Momentum in Collisions

You already know momentum is conserved in a collision. What AS glossed over is the mechanism — the actual push, over the actual time, that changes one object's momentum into another's. That push-over-time is called impulse, and it's the sharpest tool for reasoning about crashes, catches and cushioning.

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From Newton's second law to impulse

At AS you met Newton's second law as $F=ma$. That form is only the whole story when mass stays constant — fine for a trolley or a falling ball, but not for a rocket burning fuel or a raindrop picking up mass. The more fundamental statement is written in terms of momentum, $p=mv$: the resultant force on an object equals the rate of change of its momentum:

$F = \dfrac{\Delta p}{\Delta t}$

When mass is constant, $\Delta p = m\Delta v$, and dividing by $\Delta t$ recovers $F=ma$ exactly — so nothing from AS is overturned, just generalised. Rearranging gives a new and useful quantity:

$F\Delta t = \Delta p$

The left-hand side, force multiplied by the time it acts, is called the impulse, symbol $I$. Impulse is a vector, with the same direction as the force (and the resulting momentum change), with units of newton-seconds (N s) — which, since $1\,\text{N} = 1\,\text{kg}\,\text{m}\,\text{s}^{-2}$, is identical to the kg m s⁻¹ you'd get from $\Delta p$ directly.

$I = F\Delta t = \Delta p = m\Delta v$

Impulse is a change, not a force

It's tempting to think of impulse as "a kind of force," but it isn't — it's the total effect a force has once you account for how long it acts. A small force acting for a long time can deliver exactly the same impulse, and the same change in momentum, as a large force acting briefly. This is why impulse is the right tool for comparing a slow, cushioned stop against a sudden, violent one: the momentum change is identical either way, but the force involved is very different.

Force-time graphs and the area underneath

Real collisions rarely involve a constant force — think of a foot striking a football, or a car hitting a barrier: the force ramps up from zero, peaks, and falls back over a few milliseconds. $I=F\Delta t$ only applies directly when $F$ is constant, but the idea generalises neatly to a varying force using a force-time graph.

DerivationImpulse as the area under a force-time graph

Split the total time of contact into a large number of very short intervals, each of duration $\delta t$, small enough that the force $F$ is effectively constant over each one.

Over each tiny interval, the impulse delivered is $F\,\delta t$ — exactly the area of a thin vertical strip under the force-time graph at that instant.

Height of the strip is $F$, width is $\delta t$, so strip area = $F\,\delta t$.

Summing the impulse from every strip across the whole contact time gives the total impulse: $I = \sum F\,\delta t$.

In the limit of infinitesimally thin strips, this sum becomes the exact area under the curve, i.e. $I=\int F\,dt$.

So for any force-time graph, constant or varying, the total impulse — and hence the total change in momentum — equals the area enclosed between the curve and the time axis.

For a constant force this area is just a rectangle, $F\times\Delta t$, recovering the simple formula as a special case.

▪ derived

This is genuinely useful on exam graphs: even when a force-time curve is an irregular hump with no simple algebraic shape, you can still find the impulse — and hence the change in momentum — by counting squares or using the trapezium rule to estimate the enclosed area.

Why impulse explains crumple zones and cushioning

Rearrange $I=F\Delta t=\Delta p$ for the force: $F=\dfrac{\Delta p}{\Delta t}$. For a given collision, $\Delta p$ — say, a car going from cruising speed to rest — is fixed by the physics of the crash: mass and speed change don't care what the barrier is made of. What you can influence is $\Delta t$, the time over which that momentum change happens.

Stretch $\Delta t$ out — a longer contact time — and $F$ falls proportionally, since $\Delta p$ stays the same. This is exactly what a crumple zone, an airbag, a cricketer's "giving" hands catching a fast ball, or a gymnast's cushioned landing mat all do: none change how much momentum has to be lost, they simply spread that loss over a longer time, capping the peak force and reducing the risk of injury or damage.

Worked example

A 900 kg car travelling at 15 m s⁻¹ collides with a wall and is brought to rest. Case A: a rigid, non-crumpling bumper brings it to rest in 0.05 s. Case B: a crumple zone extends this to 0.4 s.

$\Delta p = m\Delta v = 900\times15 = 1.35\times10^{4}\,\text{kg m s}^{-1}$ in both cases — the impulse required is identical.

Case A: $F=\dfrac{1.35\times10^4}{0.05}=2.7\times10^5\,\text{N}$. Case B: $F=\dfrac{1.35\times10^4}{0.4}=3.375\times10^4\,\text{N}$ — roughly eight times smaller, for exactly the same change in momentum, purely because the collision was stretched out over eight times as long.

Impulse in two-body collisions

In any collision between two isolated objects, Newton's third law guarantees the force each exerts on the other is equal and opposite at every instant, and since both act for the same contact time $\Delta t$, the impulses delivered to the two objects are also equal and opposite. That's the direct mechanical reason total momentum is conserved in a collision — not a separate law bolted on top of Newton's laws, but a direct consequence of them.

$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$

where $u$ is velocity before the collision and $v$ after. This is the same conservation of momentum equation from AS — impulse just gives the underlying reason it must be true, and a way of finding the actual contact force if you know (or can estimate) the collision time.

Key takeaways

Impulse $I = F\Delta t = \Delta p$, a vector measured in N s (equivalent to kg m s⁻¹).
Newton's second law in its general form is $F=\dfrac{\Delta p}{\Delta t}$, which reduces to $F=ma$ only when mass is constant.
For any force-time graph, the impulse delivered equals the area enclosed between the graph and the time axis — this holds even when the force varies.
For a fixed change in momentum, stretching the collision time $\Delta t$ reduces the peak force $F$ proportionally — the principle behind crumple zones, airbags and cushioned landings.
Newton's third law means the impulses on two colliding bodies are equal and opposite, which is the mechanical origin of conservation of momentum.

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