Unit 3 · Mechanics (Paper 4)
The Complete Cambridge International AS & A Level Mathematics (9709) Guide · Cambridge International AS & A Level Mathematics (9709)Unit 3 · Mechanics (Paper 4)
- The Complete Cambridge International AS & A Level Mathematics (9709) Guide
- Cambridge International AS & A Level Mathematics (9709)
- 5 sections
Cambridge International does not publish a per-unit weighting for this exam. Exam structure (official): AS Level: Pure Mathematics 1 (110 min, 60%) · Pure Mathematics 2 (75 min, 40%) · Mechanics (75 min, 40%) · Probability & Statistics 1 (75 min, 40%); A Level: Pure Mathematics 1 (110 min, 30%) · Pure Mathematics 3 (110 min, 30%) · Mechanics (75 min, 20%) · Probability & Statistics 1 (75 min, 20%) · Probability & Statistics 2 (75 min, 20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. Units and directions turn arithmetic into a physical answer.
- How Cambridge International AS & A Level Mathematics (9709) assesses this Cambridge International publishes no per-unit percentage for Cambridge International A Level Mathematics; Pure Mathematics 1 accounts for 60% of the AS Level total and 30% of the A Level total. · AS Level: Pure Mathematics 1 (110 min, 60%) · Pure Mathematics 2 (75 min, 40%) · Mechanics (75 min, 40%) · Probability & Statistics 1 (75 min, 40%) | A Level: Pure Mathematics 1 (110 min, 30%) · Pure Mathematics 3 (110 min, 30%) · Mechanics (75 min, 20%) · Probability & Statistics 1 (75 min, 20%) · Probability & Statistics 2 (75 min, 20%) · show the model with free-body diagram with resolved force components, velocity-time information and constant-acceleration equations, momentum or energy ledger with units and directions
- Key skills Forces and Equilibrium: Resolving Forces, Contact Forces and Friction, Kinematics of Motion in a Straight Line: Graphs, Constant Acceleration and Calculus, Momentum: Conservation of Linear Momentum in Direct Impacts, Newton's Laws of Motion: Inclined Planes and Connected Particles, Energy, Work and Power: Work Done, Kinetic and Potential Energy, Conservation of Energy
- How to study for Unit 3 This page turns free-body diagram with resolved force components, velocity-time information and constant-acceleration equations, momentum or energy ledger with units and directions into one route: Draw or describe the system and write the direction and force model before forming an equation..
- The organizing decision Define the body or connected system, choose a positive direction, and decide whether forces, motion equations, momentum, or energy gives the shortest complete route.
What Cambridge International AS & A Level Mathematics (9709) Unit 3 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Forces and Equilibrium: Resolving Forces, Contact Forces and Friction
resultant force; limiting friction; equilibrium02Kinematics of Motion in a Straight Line: Graphs, Constant Acceleration and Calculus
equilibrium; acceleration; tension03Momentum: Conservation of Linear Momentum in Direct Impacts
tension; momentum; impulse04Newton's Laws of Motion: Inclined Planes and Connected Particles
impulse; work-energy relation; power05Energy, Work and Power: Work Done, Kinetic and Potential Energy, Conservation of Energy
power; resultant force; limiting frictionCambridge International does not publish a per-unit weighting for this exam.
Official unit name and weighting: Cambridge International syllabus.
Forces and motion
Connect the published share to the unit model
Units and directions turn arithmetic into a physical answer.
For forces and motion, the following pages use newly written examples and data. The forces and motion scoring comparisons are learning aids based on the published syllabus and the general requirement to show necessary working, not official mark schemes.
Before work on forces and motion, identify the domain, units, accuracy instruction, and event boundary. Preserve exact values in forces and motion until a final approximation is requested.
The decision that organizes this unit
Define the system and choose the route before calculating
Define the body or connected system, choose a positive direction, and decide whether forces, motion equations, momentum, or energy gives the shortest complete route.
Draw or describe the system and write the direction and force model before forming an equation.
Mechanism route and repair branches
Relationships to preserve
- Resolving forces along chosen axes links a force diagram to equilibrium or acceleration.
- Connected particles share a compatible acceleration magnitude while tension acts in opposite directions on the two bodies.
- Work done by the engine supplies changes in kinetic and gravitational potential energy together with work against resistance.
Representations to read
- free-body diagram with resolved force components
- velocity-time information and constant-acceleration equations
- momentum or energy ledger with units and directions
Branches to reject
- changing the sign convention between equations
- treating connected particles as if their accelerations were unrelated
- using path length for gravitational height or omitting resistive work
| Key concept | Why it's hard | What scores |
|---|---|---|
| Force balance | Friction is responsive up to a limit | Compares required and limiting friction |
| Changing motion | Integration constants carry initial data | Uses velocity and displacement with units |
| System methods | Momentum and energy have different conditions | States why the chosen conservation law applies |
How Cambridge International AS & A Level Mathematics (9709) assesses Mechanics (Paper 4)
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Forces and Equilibrium: Resolving Forces, Contact Forces and Friction | resultant force; limiting friction; equilibrium | changing the sign convention between equations |
| Kinematics of Motion in a Straight Line: Graphs, Constant Acceleration and Calculus | equilibrium; acceleration; tension | treating connected particles as if their accelerations were unrelated |
| Momentum: Conservation of Linear Momentum in Direct Impacts | tension; momentum; impulse | using path length for gravitational height or omitting resistive work |
| Newton's Laws of Motion: Inclined Planes and Connected Particles | impulse; work-energy relation; power | changing the sign convention between equations |
| Energy, Work and Power: Work Done, Kinetic and Potential Energy, Conservation of Energy | power; resultant force; limiting friction | treating connected particles as if their accelerations were unrelated |
Equilibrium with friction
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Equilibrium with friction
Take the acceleration due to gravity to be 9.8 m s⁻².
A crate of mass 6 kg rests on a rough horizontal floor. The coefficient of friction between the crate and the floor is 0.35. A horizontal force of 18 N is applied to the crate. Determine whether the crate moves, and state the magnitude and direction of the frictional force.
[3]Show all necessary working.
A. R = 58.8 N and μR = 20.58 N; since 18 < 20.58, the crate stays at rest and friction is 18 N opposite the pull.
✅ Correct! 😊 Correct. Static friction supplies the required 18 N because this is below its limiting value of 20.58 N.
B. R = 58.8 N, so friction is always μR = 20.58 N and the crate accelerates opposite the pull.
❌ Not quite 😭 This treats limiting friction as the actual friction. Static friction adjusts to 18 N here, so the horizontal forces balance.
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C. Since 18 N is non-zero, the crate moves in the direction of the applied force and friction is 20.58 N.
❌ Not quite 😭 A non-zero applied force does not guarantee motion. The available static friction exceeds the force required for equilibrium.
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D. The crate stays at rest because 18 < 58.8, and the frictional force is 0.35 N.
❌ Not quite 😭 This compares the pull with the normal reaction rather than the limiting friction and mistakes the coefficient for a force.
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Full explanation
- Step 1Vertical equilibrium gives R = 6 × 9.8 = 58.8 N.
- Step 2The limiting friction is μR = 0.35 × 58.8 = 20.58 N.
- Step 3Only 18 N of friction is required for equilibrium, and 18 < 20.58, so static friction can balance the pull.
- Step 4The crate remains at rest; the frictional force is 18 N opposite the applied force.
Practice 2: Connected particles
Take the acceleration due to gravity to be 9.8 m s⁻².
Particles P and Q, of masses 3 kg and 2 kg respectively, are connected by a light inextensible string passing over a smooth pulley. P hangs vertically and Q rests on a smooth horizontal table. The system is released from rest. Find the acceleration of the particles and the tension in the string.
[5]Show all necessary working.
A. For P: 29.4 − T = 3a; for Q: T = 2a.
✅ Correct! 😊 Correct. P accelerates downward while Q accelerates toward the pulley, and the inextensible string gives both particles the same acceleration magnitude.
B. For P: 29.4 + T = 3a; for Q: T = 2a.
❌ Not quite 😭 This puts tension in the same direction as P's weight. Tension acts upward on the hanging particle, opposing its downward motion.
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C. For P: 29.4 − T = 3a; for Q: T = 2b, where a and b are unrelated.
❌ Not quite 😭 The string is inextensible, so the connected particles cannot have unrelated acceleration magnitudes while the string remains taut.
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D. For the whole system: 29.4 = 3a, and T = 2a.
❌ Not quite 😭 This applies the hanging weight only to P's mass when modeling the whole system. The external driving force accelerates the total mass of 5 kg.
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Full explanation
- Step 1Choose downward for P and toward the pulley for Q as positive; both particles have acceleration magnitude a.
- Step 2Newton's second law gives 29.4 − T = 3a for P and T = 2a for Q.
- Step 3Adding eliminates T: 29.4 = 5a, so a = 5.88 m s⁻².
- Step 4Substitute into T = 2a to obtain T = 11.76 N.
Key terms for Unit 3: Mechanics (Paper 4)
Models, uses, and boundaries
- Force, momentum, and kinetic-energy relations organize the mechanics methods in this unit
- Resolve the applied forces along the chosen direction and use equilibrium or force equals mass times acceleration as appropriate. Force, momentum, and kinetic-energy relations organize the mechanics methods in this unit.
- Resolving forces and applying limiting friction produce the displayed equilibrium condition
- Use constant-acceleration relationships to connect displacement, velocity, acceleration, and time. Resolving forces and applying limiting friction produce the displayed equilibrium condition.
- The constant-acceleration displacement and momentum equation describe the motion before and during impact
- Conserve momentum across the stated interaction while keeping the chosen direction signs visible. The constant-acceleration displacement and momentum equation describe the motion before and during impact.
- The two equations of motion share one acceleration and determine the tension
- Write one motion equation for each connected particle and eliminate the shared tension. The two equations of motion share one acceleration and determine the tension.
- The work total combines the kinetic-energy change, gain in potential energy, and resistance term
- Add the kinetic-energy gain, gravitational-energy gain, and work against resistance to obtain the engine's work. The work total combines the kinetic-energy change, gain in potential energy, and resistance term.
Cambridge International AS & A Level Mathematics (9709) Unit 3 FAQ
How much of Cambridge International AS & A Level Mathematics (9709) does Unit 3 carry?
Cambridge International does not publish a per-unit weighting for this exam.
What is the first move on a Mechanics (Paper 4) problem?
Draw or describe the system and write the direction and force model before forming an equation.
Which relationships should I preserve?
Resolving forces along chosen axes links a force diagram to equilibrium or acceleration. Connected particles share a compatible acceleration magnitude while tension acts in opposite directions on the two bodies. Work done by the engine supplies changes in kinetic and gravitational potential energy together with work against resistance.
Which representations should I practice?
Practice moving among free-body diagram with resolved force components, velocity-time information and constant-acceleration equations, momentum or energy ledger with units and directions.
What error should I check before submitting an answer?
Check for changing the sign convention between equations; treating connected particles as if their accelerations were unrelated; using path length for gravitational height or omitting resistive work.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Equilibrium with friction
- Kinematics and impact
- Connected particles
- Energy and power
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related Cambridge International AS & A Level Mathematics (9709) unit guides
Cambridge International AS & A Level Mathematics (9709) Exam Guide & Review
The whole exam and its official unit sequence.01Pure Mathematics 1 (Paper 1)
Cambridge International does not publish a per-unit weighting for this exam.02Pure Mathematics 3 (Paper 3, A Level)
Cambridge International does not publish a per-unit weighting for this exam.04Probability & Statistics 1 (Paper 5)
Cambridge International does not publish a per-unit weighting for this exam.05Probability & Statistics 2 (Paper 6, A Level)
Cambridge International does not publish a per-unit weighting for this exam.How to study Cambridge International AS & A Level Mathematics (9709) Unit 3
Start with the organizing decision
Before solving, restate the decision in operational terms: Define the body or connected system, choose a positive direction, and decide whether forces, motion equations, momentum, or energy gives the shortest complete route.. Your first written move should be to Draw or describe the system and write the direction and force model before forming an equation..
Practice the same idea in several representations
Rotate through free-body diagram with resolved force components, velocity-time information and constant-acceleration equations, momentum or energy ledger with units and directions. Use each representation to practice Forces and Equilibrium: Resolving Forces, Contact Forces and Friction, Kinematics of Motion in a Straight Line: Graphs, Constant Acceleration and Calculus, Momentum: Conservation of Linear Momentum in Direct Impacts, Newton's Laws of Motion: Inclined Planes and Connected Particles, Energy, Work and Power: Work Done, Kinetic and Potential Energy, Conservation of Energy, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for changing the sign convention between equations; treating connected particles as if their accelerations were unrelated; using path length for gravitational height or omitting resistive work. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official Cambridge International syllabus for the 2026-2027 administration.
Where to go next
Return to the Cambridge International AS & A Level Mathematics (9709) to see every unit and plan your revision.