Mini-Lesson

Impulse and momentum

This mini-lesson covers momentum p = mv, impulse I = Ft = mv − mu, and the conservation of momentum in collisions, coalescence (bodies joining) and explosions. Take g = 9.8 m s⁻² where needed.

Where this sits: Unit A2 2: Applied Mathematics — momentum and impulse. ⚠️ This is a genuine CCEA difference: impulse and momentum are part of CCEA A-level Mathematics, whereas on AQA, Edexcel and OCR they sit in Further Mathematics. Treat them as core content.

Work through each screen, answer the questions as you go (some are reasoning, some are calculations) and collect ⭐ stars. Press Start when you are ready.

Momentum

Momentum p = mv

momentum = mass × velocity, p = mvunits: kg m s⁻¹ (equivalently N s) · momentum is a VECTOR — direction matters

CCEA-specific: impulse and momentum are examined in Unit A2 2 (Applied) at CCEA. Students following AQA/Edexcel/OCR A-level Mathematics do not meet this content — it is Further Maths for them. So do not be surprised if a textbook aimed at another board omits it.

Worked example

A 0.5 kg ball moves at 12 m/s.

p = mv = 0.5 × 12 = 6 kg m s⁻¹

Signs: choose a positive direction first. A body moving the other way has negative velocity and therefore negative momentum.

Calculate

Momentum

1A ball of mass 0.5 kg moves at 12 m/s. Find its momentum.

kg m/s

Hint: p = mv = 0.5 × 12.

Impulse

The impulse–momentum principle

impulse I = Ft = mv − muimpulse = change in momentum · units N s (= kg m s⁻¹) · also a VECTOR

A constant force F acting for time t delivers impulse Ft. That impulse is the change in momentum — which is why a longer contact time (a follow-through, a crumple zone) reduces the force for the same change in momentum.

Worked example — a ball rebounding

A 0.15 kg ball hits a wall at 20 m/s and rebounds at 25 m/s.

Take the rebound direction as positive: u = −20 m/s, v = +25 m/s.

I = m(v − u) = 0.15(25 − (−20)) = 0.15 × 45 = 6.75 N s

If the contact lasts 0.05 s, the average force is F = I/t = 6.75 ÷ 0.05 = 135 N

The sign is the whole question. Using 25 − 20 = 5 instead of 45 is the classic error — the ball reversed, so the velocities have opposite signs.

Calculate

Impulse on a rebounding ball

2A 0.15 kg ball strikes a wall at 20 m/s and rebounds at 25 m/s. Taking the rebound direction as positive, find the magnitude of the impulse.

N s

Hint: u = −20, v = +25. I = m(v − u) = 0.15 × (25 − (−20)) = 0.15 × 45.

Calculate

Average force

3The impulse in the last question is 6.75 N s and the contact lasts 0.05 s. Find the average force on the ball.

N

Hint: I = Ft, so F = I ÷ t = 6.75 ÷ 0.05.

Conservation

Conservation of linear momentum

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂in a collision with no external force, TOTAL momentum before = TOTAL momentum after

Momentum is conserved because the two bodies exert equal and opposite forces on each other (Newton's third law) for the same time — so the impulses cancel.

Worked example — a collision where the bodies separate

A 4 kg body at 6 m/s hits a stationary 2 kg body. After the collision the 4 kg body still moves forwards at 3 m/s. Find the speed of the 2 kg body.

Before: 4(6) + 2(0) = 24 · After: 4(3) + 2v = 12 + 2v

24 = 12 + 2v ⇒ v = 6 m/s in the original direction

Kinetic energy is NOT generally conserved in a collision (some is lost as heat and sound) — but momentum always is, provided no external impulse acts.

Calculate

Coalescence

4A 2 kg trolley moving at 5 m/s collides with a stationary 3 kg trolley and they join together. Find their common velocity afterwards.

m/s

Hint: Momentum before = 2 × 5 + 3 × 0 = 10 kg m/s. After they coalesce the combined mass is 5 kg, so 10 = 5v.

Calculate

Collision with separation

5A 4 kg body moving at 6 m/s strikes a stationary 2 kg body. After impact the 4 kg body continues forwards at 3 m/s. Find the speed of the 2 kg body after impact.

m/s

Hint: Momentum before = 4 × 6 = 24. Momentum after = 4 × 3 + 2v = 12 + 2v. So 24 = 12 + 2v.

Explosions

Explosions and recoil

An explosion is a collision in reverse: the bodies start together (or at rest) and fly apart. Momentum is still conserved, so if the system starts at rest:

0 = m₁v₁ + m₂v₂ ⇒ m₁v₁ = −m₂v₂the two bodies move in OPPOSITE directions with equal and opposite momenta

Worked example — recoil

A 3 kg rifle fires a 0.02 kg bullet at 300 m/s. Total momentum before = 0.

0 = 0.02(300) + 3v ⇒ 3v = −6 ⇒ v = −2 m/s

The rifle recoils at 2 m/s in the opposite direction to the bullet.

Quick check

Units of impulse

?Which of these is a correct unit for impulse?

Sort it

Momentum, impulse or neither?

Tap an expression, then tap what it represents.

🏃 Momentum

💥 Impulse

🚫 Neither

Quick check

Why momentum is conserved

?In a collision, why is total momentum conserved?

Match it

Match the momentum idea

Tap the term on the left, then its meaning on the right.

Statement

Answer

Quick check

Crumple zones

?Why does a car's crumple zone reduce the force on the passengers in a crash?

Quick check

Direction matters

?A 2 kg ball moving right at 3 m/s collides head-on with a 1 kg ball moving left at 6 m/s. What is the total momentum before impact (taking right as positive)?

Examiner traps

Impulse and momentum pitfalls

Signs on a rebound. The velocity reverses, so I = m(v − u) with u negative — the sizes ADD.
Impulse is a change, not a state: I = mv − mu, never just mv.
Momentum is conserved; kinetic energy generally is not. Do not equate KE before and after.
Coalescence: after joining, the combined mass moves with ONE velocity — use (m₁ + m₂)v.
Explosion from rest: total momentum stays zero, so the fragments have equal and opposite momenta.
Units: momentum kg m s⁻¹, impulse N s — numerically the same thing.

Recap

The big ideas to know

Momentum: p = mv (kg m s⁻¹) — a VECTOR, so fix a positive direction first

Impulse: I = Ft = mv − mu (N s) — impulse IS the change in momentum

Rebounds: the velocity reverses sign, so I = m(v − (−u)) = m(v + u) in magnitude

Conservation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ when no external impulse acts

Coalescence: one combined mass, one common velocity · Explosion: equal and opposite momenta from rest

⚠️ CCEA-specific: this content is A-level (Unit A2 2) at CCEA, but Further Maths on other boards

Momentum completes the CCEA mechanics course: kinematics, forces, moments and impulse. Press Finish to see your score.

🏆

Mini-lesson complete!

⭐⭐⭐

You have worked through Impulse and momentum for CCEA GCE Mathematics. 🎉

Your stars: 0 / 0

Next: test yourself in the Evaluate stage Confidence Quiz, then lock it in with Verify.

📣 Smashed it? Share your score

Challenge a mate to beat your stars, or show a parent how you got on.

→ Back to all subjects