🧠 Impulse and Momentum: Collisions, Rebounds & Conservation

Right — collisions. Particles smash, bounce, reverse, stick together, pass momentum like a secret handshake — and students instantly panic about signs, directions, elastic vs inelastic, impulses, and whether momentum magically disappears (it doesn’t). We’re going to walk through this like we’re sketching things live on a whiteboard together — slightly messy, pauses included, no polished textbook tone. Just real thinking.

Picture two balls colliding. One slows, the other speeds up. Total momentum before = total momentum after — always, provided no external forces interfere. The difficulty isn’t the law — it’s tracking direction and sign choices without letting your brain quietly flip one. If you get that sorted early, collision problems become surprisingly calm. This is the moment where mechanics starts feeling like part of your A Level Maths understanding for mocks rather than chaos with subscripts.

🔙 Previous topic:

Our work on Work, Energy & Power: Efficiency, Resistance & Real Exam Questions sets the stage nicely here, because once you understand how energy flows through a system, it’s much easier to make sense of the sudden momentum changes that happen during collisions and rebounds.

📚 Where Collisions Appear in Real Exam Papers

Collisions show up everywhere — straight line impacts, smooth surfaces, two-particle rebound questions, impulse arguments, coefficient of restitution, velocity reversal after impact. Examiners love when students set up momentum equations confidently, then apply rebound logic without freezing.

📐 Setting Up the Collision Scenario

Two particles of masses $m_1$ and $m_2$ move along a straight line with initial velocities $u_1$ and $u_2$ . They collide directly. After impact, their velocities become $v_1$ and $v_2$ . Prove that total momentum is conserved and calculate the impulse during the collision.

🧩 Core Ideas Behind Momentum & Impulse

⚖️ Momentum — A Directional Quantity Students Often Underrate

Momentum is directional. If one object goes right and one goes left, they can cancel. That’s where sign awareness matters.

We write momentum as something like ( $p = mv$ )

For a two-particle system before collision:

We might form ( $m_1u_1 + m_2u_2$ )

After collision:

We write ( $m_1v_1 + m_2v_2$ )

Equate them for conservation:

We might form ( $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ )

That one line drives 70% of collision questions.

Pause a second. It’s not a formula — it’s a balance. Before and after must weigh the same.

🎯 Impulse — The Hidden Story Behind Sudden Changes

Impulse is just a change in momentum. Nothing mystical.

We write ( $I = \Delta p = mv – mu$ )

Or in force form:

We might use ( $I = F \times t$ )

Which one you use depends on the data given. If force + time is known → impulse first. If velocities are known → momentum difference first.

Impulse tells you how violently an object was hit. Hard punch, fast swing, small collision time — large impulse.

🔄 Elastic vs Inelastic — What Actually Stays the Same?

People memorise definitions. You don’t need to. Think of energy.

Elastic collision: kinetic energy is conserved
Inelastic: kinetic energy is lost somewhere (heat, sound, deformation)
Perfectly inelastic: particles stick together, move with one final shared velocity

Momentum always holds. Kinetic energy sometimes holds.

If elastic:

We write ( $\tfrac12 m_1u_1^2 + \tfrac12 m_2u_2^2 = \tfrac12 m_1v_1^2 + \tfrac12 m_2v_2^2$ )

If inelastic, KE decreases but the momentum equation still works.

This is the mid-blog moment where collision questions start feeling predictable — the kind of confidence that comes from A Level Maths revision that sticks, not simple memorisation.

🧮 Coefficient of Restitution — Measuring the “Bounce”

Now the part examiners love.

Coefficient of restitution measures “bounciness”.

We write something like:

( $e = \frac{\text{speed of separation}}{\text{speed of approach}}$ )

Range:

(e = 1) → perfectly elastic (no KE lost)
(0 < e < 1) → partially elastic (KE lost)
(e = 0) → perfectly inelastic (objects stick)

If two objects rebound, separation speeds take signs into account. Watch that carefully — one arrow may flip direction.

📝 Worked Example — Keeping Signs Straight and Equations Calm

Two particles A and B:

\begin{aligned} m_1 &= 2\,\text{kg}, \quad u_1 = 5\,\text{m s}^{-1}\text{ (right)}\\ m_2 &= 3\,\text{kg}, \quad u_2 = 1\,\text{m s}^{-1}\text{ (right)}\\ v_1 &= 2\,\text{m s}^{-1}\text{ (right)} \end{aligned}

After the collision, find $v_2$ and the impulse on each particle.

Momentum:

We write ( $2(5) + 3(1) = 2(2) + 3v_2$ )

So:

( $13 = 4 + 3v_2$ )
( $v_2 = 3\text{ m/s}$ )

Impulse on A:

Negative means the impulse is opposite to the direction of motion. B receives $+6\,\text{N s}$ , equal in magnitude and opposite in direction.

Negative means the impulse is opposite to the direction of motion. B receives $+6\,\mathrm{N\,s}$ , equal in magnitude and opposite in direction.

I like pausing here — this is Newton’s Third Law humming underneath.

🔁 Rebounds — Handling the Velocity Flip Without Panic

If object reverses direction:

Incoming (u = +3) m/s
Rebound (v = -2) m/s

Impulse becomes:

We write ( $I = m(-2 - 3) = -5m$ )

Large magnitude — big direction change. That’s what rebound means mathematically.

Students slip when they forget to flip the velocity — treat signs like traffic laws.

⚠️ Exam Pitfalls — The Ones That Trap Even Strong Students

Using speeds instead of velocities — signs matter
Forgetting impulse = momentum change, not force
Treating KE like always conserved — only elastic
Swapping “approach” and “separation” in (e)
Momentum must balance — external forces ruin it

One LaTeX snippet to keep clear:

Momentum = conserved
Kinetic energy = sometimes conserved

Different sentences — never swap them.

🌍 Real-World Link — Where Momentum Shows Up Constantly

Billiard balls, rugby tackles, cars crumpling in slow-mo crash tests, tennis balls rebounding off court surfaces — impulse and momentum are everywhere. Soft collisions reduce rebound (energy lost to shape change). Hard collisions bounce cleanly (energy retained). The maths just models the physics you’ve felt your whole life.

🚀 Next Steps — Taking Collision Confidence Further

If you want collision modelling — impulse jumps, rebound velocity, restitution, KE comparisons — to feel instinctive rather than lucky, the A Level Maths Revision Course that explains everything trains collision intuition with diagrams, worked exam setups, velocity sign tracking and KE-vs-momentum thinking until it becomes second nature.

📏 Recap Table

Momentum: ( $p = mv$ )
Impulse: ( $I = \Delta p = mv – mu$ )
Conservation: ( $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ )
KE conserved only if elastic
Coefficient of restitution controls rebound severity

Author Bio – S. Mahandru

I teach collisions like slow-motion replays — arrows flipping, impulses transferring, momentum balancing like scales. Once direction choice stops feeling fragile, the whole topic becomes almost satisfying.

🧭 Next topic:

After getting comfortable with collisions and momentum jumps, the natural step forward is How Calculus Explains Motion Through Velocity and Acceleration, where motion stops being constant and you switch to integration to track how velocity evolves.

❓ Quick FAQs

Why is momentum always conserved but kinetic energy sometimes isn’t?

Momentum behaves like a kind of “currency” that can only change if something outside the system pushes or pulls on it. In a two-particle collision on a smooth surface, no external forces interfere, so the total momentum before and after must balance — even if the particles smack into each other in dramatic ways. Kinetic energy, though, is far more delicate. It can drain into sound, heat, deformation, vibration, or the tiny flexing of materials during impact. Nothing illegal there — physics just redistributes the energy into forms we’re not tracking.

Students sometimes assume KE “should” stay the same because they’ve only seen perfect textbook bounces, but real collisions almost always bleed energy away. The key idea: momentum cares about the whole system; kinetic energy cares about how the system stores energy. Once you stop expecting both to behave identically, conservation laws feel much less mysterious. And honestly, once that clicks, most collision questions untangle themselves without much drama.

Does impulse always mean the object changes direction?

Not at all — impulse simply measures how much momentum changes, and that includes increases, decreases, or full reversals. Sometimes an object slows down but keeps moving in the same direction; the impulse is smaller, and the sign stays predictable. Other times, especially in rebound situations, the velocity flips sign, and suddenly the impulse becomes a large negative or positive value depending on your direction choice.

What usually confuses students isn’t the physics — it’s the bookkeeping. Forgetting to switch the sign on the rebound velocity is the classic move that breaks an entire question. Impulse only “forces a reversal” if the change in momentum is big enough compared to the object’s original motion. Think of impulse as a shove: a gentle one tweaks your speed, a big one boots you back the way you came. Once you read it like that, the maths starts feeling like a story instead of a trap.

Should I use momentum or energy in a collision question?

Momentum is non-negotiable — you always use it, because collisions are built on it. Energy is optional, and only appears when the question is specifically about elastic impact or KE comparison. Many students try to bring energy into every collision because it worked once, but that’s how you end up doing three extra lines of algebra for no gain.

Momentum gives you your structural equation: before = after. Energy, if the collision is elastic, gives you a second equation that helps you solve for unknown velocities. In inelastic cases, the KE equation doesn’t hold, so forcing it will derail everything instantly. Examiners love this distinction — they can tell in one line whether you understand the nature of the collision or whether you’re applying formulas blindly.
A good rule of thumb: momentum every time, energy only when kinetic energy is intended to survive the impact. And if the question mentions things sticking together, forget KE entirely — it’s gone.

🧠 Impulse and Momentum: Collisions, Rebounds & Conservation