Mathematics 9709 · syllabus 4.3
Cambridge A Level Mathematics 9709 — Momentum topical past papers
Linear momentum as mass times velocity, and its conservation when two particles collide directly or coalesce, all in one dimension.
- Questions
- 40
- Years
- 2020–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 28
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Momentum covers
The momentum of a particle is its mass times its velocity, mv, measured in N s or kg m s⁻¹. Because velocity has a direction, so does momentum, and in Paper 4 (section 4.3) all motion is along one straight line, so its direction is shown by a sign. When two particles collide directly and no external force acts along the line of motion, the total momentum before the collision equals the total momentum after it.
Questions give some of the masses and velocities before and after a collision and ask for the rest, or describe particles that coalesce and move on together. Some ask whether a particle's direction has been reversed, or combine a collision with a later stage of motion, such as the particles then slowing under friction. Impulse and the coefficient of restitution are not part of 9709.
Momentum links to the rest of Paper 4 through problems in several stages: kinematics or Newton's second law gives a speed just before an impact, conservation of momentum gives the speeds after it, and energy may then show how much kinetic energy the collision lost. A particle brought to rest, or one whose direction reverses, often marks the change from one stage to the next, so the sign of each velocity needs care throughout.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 4.3 Momentum
What you should be able to do
- Calculate momentum, using signs for direction
- Apply conservation of momentum to two particles colliding directly
- Find the common velocity when particles coalesce
- Decide from the sign of an answer whether a particle's direction has reversed
- Combine a collision with kinematics, Newton's laws or energy in a problem with several stages
Key ideas
- Momentum = mv, a vector; in one dimension its sign gives its direction
- Conservation of momentum in a direct collision: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
- Particles that coalesce: m₁u₁ + m₂u₂ = (m₁ + m₂)v
- Kinetic energy is generally lost in a collision, even though momentum is conserved
How Momentum has been examined
From 2022 to 2026, Momentum was on Paper 4 · Mechanics in 10 of 11 exam series, worth about 12% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 4 · Mechanics | 28 | 12% | 10 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 6.
Filed in Momentum under: Momentum.
A worked example
Particle A, of mass 0.3 kg and moving at 5 m s⁻¹, collides directly with particle B, of mass 0.2 kg, which is moving towards A at 2 m s⁻¹. After the collision B moves at 3 m s⁻¹ in the opposite direction to its original motion. Find the velocity of A after the collision.
- Take A's original direction as positive, so B's velocity is −2 m s⁻¹ before the collision and +3 m s⁻¹ after it.
- Momentum before = 0.3 × 5 + 0.2 × (−2) = 1.5 − 0.4 = 1.1 N s.
- Momentum after = 0.3v + 0.2 × 3 = 0.3v + 0.6, where v is A's velocity.
- Conservation of momentum: 0.3v + 0.6 = 1.1, so v = 0.5 / 0.3 = 5/3.
- v is positive, so A keeps moving in its original direction.
Answer: 5/3 m s⁻¹ (1.67 m s⁻¹) in A's original direction
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Giving every velocity a positive sign when particles move towards each other: choose a positive direction and make velocities the other way negative.
- Treating a negative answer as an error rather than as motion in the opposite direction to the one chosen as positive.
- Assuming kinetic energy is conserved in a collision: only momentum is, unless the question says otherwise.
- Using weight, mg, instead of mass in the momentum equation: momentum is mass × velocity.
Exam technique
- Draw 'before' and 'after' diagrams showing masses, velocities and a marked positive direction.
- Write the conservation equation in full before substituting, so the sign of each term is visible.
- State the direction of motion in words in the final answer, such as 'moving at 2 m s⁻¹ in its original direction'.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.