Mathematics 9709 · syllabus 4.4
Cambridge A Level Mathematics 9709 — Newton's laws of motion topical past papers
Using F = ma for particles under constant forces, on inclined planes, and as connected particles joined by strings over pulleys or by tow-bars.
- Questions
- 197
- Years
- 2009–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 59
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Newton's laws of motion covers
Newton's second law, resultant force = mass × acceleration, is at the centre of Paper 4 (section 4.4). For a particle of constant mass under constant forces, you resolve in the direction of motion, set the resultant equal to ma, and then use kinematics to find speeds, distances and times. The forces include weight (W = mg, with g = 10 m s⁻²), friction, tension in a light inextensible string, thrust in a rod, and any resistance the question describes.
Two settings recur. On an inclined plane, a particle may move up and then down a rough slope with different accelerations, because friction always opposes the motion. With connected particles, such as two masses joined by a string over a smooth pulley or a car towing a trailer, you write F = ma for each particle separately, or for the whole system when the internal force cancels, and solve the equations together.
The topic joins forces and equilibrium with kinematics, and questions often add a later stage: a string breaks or a particle hits the ground, after which the remaining particle moves under different forces. The work–energy principle offers another route through some of these stages. Air resistance and other resisting forces come into a question only when it says so, and every body is treated as a particle, with all its forces acting at one point.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 4.4 Newton's laws of motion
What you should be able to do
- Resolve forces along the direction of motion and apply F = ma
- Find friction on a rough horizontal or inclined surface, using F = μR when sliding
- Write separate equations of motion for connected particles and solve them together
- Find the tension in a string or the force in a tow-bar
- Handle later stages of motion, such as after a string breaks or a particle reaches the ground
Key ideas
- Resultant force = ma, in the direction of the acceleration
- W = mg, with g = 10 m s⁻² in this paper
- A sliding particle has friction F = μR, opposing its motion
- A light inextensible string has the same tension throughout, including over a smooth pulley, and the particles it joins share the same speed and size of acceleration
How Newton's laws of motion has been examined
From 2022 to 2026, Newton's laws of motion was on Paper 4 · Mechanics in 11 of 11 exam series, worth about 30% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 4 · Mechanics | 59 | 30% | 11 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: 8.
Filed in Momentum under: Newton's Laws of Motion.
A worked example
Particle A of mass 4 kg lies on a rough horizontal table, where the coefficient of friction is 0.25. It is joined by a light inextensible string over a smooth pulley at the table's edge to particle B of mass 2 kg, which hangs freely. The system is released from rest; find the acceleration of the particles and the tension in the string.
- The weights are 40 N for A and 20 N for B. A slides, so the friction on A is μR = 0.25 × 40 = 10 N, opposing its motion.
- For B, moving down: 20 − T = 2a.
- For A, moving towards the pulley: T − 10 = 4a.
- Add the equations to eliminate T: 10 = 6a, so a = 5/3 = 1.67 m s⁻².
- Then T = 10 + 4 × 5/3 = 50/3 = 16.7 N. Check with B: 20 − 50/3 = 10/3 = 2 × 5/3.
Answer: Acceleration 1.67 m s⁻² (5/3); tension 16.7 N (50/3)
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Using the whole system's mass in one particle's equation: each particle's equation uses only the forces on that particle and its own mass.
- Keeping the same acceleration and friction direction when a particle moves up and then down a rough slope: friction reverses when the motion reverses.
- Leaving out the weight component along a slope, or using the full weight mg instead of mg sin α.
- Assuming the tension equals the weight of a hanging particle while it accelerates: that is only true in equilibrium.
Exam technique
- Draw a separate force diagram for each particle and mark the direction of acceleration on it.
- Write 'resultant force = ma' for each particle in its own direction of motion, then add the equations to eliminate the tension.
- Carry the acceleration into a constant-acceleration formula for any later distance or time, treating each stage of the motion separately.
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.