A Level Mathematics: Mechanics Practice Questions (Cambridge 9709 Paper 4)

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs — Cambridge International holds copyright in its own papers. Use these alongside the official past papers available from your board.

Take g = 10 m s⁻². Each question practises a skill tested in the June 2025 Paper 4 series. After each answer there is a tip and the real question to try next.

---

Questions

1. A box is pushed in a straight line along horizontal ground by a force of 40 N acting at 30° above the horizontal. The box moves 15 m in 6 seconds at constant speed. (a) Find the speed of the box. (b) Find the work done by the 40 N force. (c) Find the power of the 40 N force. [4]

2. Particles P and Q, of masses 0.3 kg and 0.2 kg, move in the same direction along a straight line on a smooth horizontal surface. P moves at 4 m s⁻¹ and Q moves at 1.5 m s⁻¹ ahead of it. P catches up with Q and the two particles coalesce. Find the kinetic energy lost in the collision. [4]

3. A car passes a point A at 12 m s⁻¹. It accelerates uniformly at 0.5 m s⁻² for 20 s, then decelerates uniformly at 0.4 m s⁻² until it stops at a point B. Find the distance AB. [4]

4. Particle A, of mass 2 kg, is on a rough horizontal table. It is attached by a light inextensible string, which passes over a small smooth pulley at the edge of the table, to particle B, of mass 3 kg, which hangs freely. The system is released from rest and B accelerates downwards at 4 m s⁻². Find the tension in the string and the coefficient of friction between A and the table. [4]

5. A particle P moves in a straight line. Its velocity t seconds after leaving a point O is v = 6t − t² + 7 m s⁻¹, for 0 ≤ t ≤ 5. (a) Find the maximum velocity of P. (b) Find the distance of P from O when it reaches this maximum velocity. [4]

6. A particle of mass 5 kg rests on a rough plane inclined at 30° to the horizontal. The coefficient of friction is 0.2. A force P N acting up the plane, parallel to a line of greatest slope, keeps the particle from sliding down. Find the least possible value of P. [4]

7. A particle of mass 2 kg is projected at 6 m s⁻¹ up a line of greatest slope of a rough plane inclined at 30° to the horizontal. The friction force has magnitude 5 N. Using an energy method: (a) find the distance the particle travels up the plane before it comes to rest (b) find its speed when it returns to its starting point. [6]

---

Answers

1. (a) 15 ÷ 6 = 2.5 m s⁻¹ [1]. (b) Work done = 40 cos 30° × 15 [1] = 520 J (3 s.f.) [1]. (c) Power = 40 cos 30° × 2.5 = 86.6 W [1].

Tip: only the component of the force in the direction of motion does work.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 42, Question 1.

2. Momentum: 0.3 × 4 + 0.2 × 1.5 = 0.5v [1], so v = 3 m s⁻¹ [1]. KE before = ½(0.3)(16) + ½(0.2)(2.25) = 2.625 J; KE after = ½(0.5)(9) = 2.25 J [1]. KE lost = 0.375 J [1].

Tip: decide one positive direction first. Here both particles move the same way, so both velocities are positive.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 42, Question 2.

3. After 20 s: v = 12 + 0.5 × 20 = 22 m s⁻¹ [1]. Distance while accelerating = 12 × 20 + ½ × 0.5 × 20² = 340 m [1]. Distance while decelerating: 0 = 22² − 2 × 0.4 × s, so s = 605 m [1]. AB = 945 m [1].

Tip: sketch the velocity–time graph first; the distance is the area under it.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 42, Question 4.

4. For B: 30 − T = 3 × 4, so T = 18 N [2]. For A: 18 − F = 2 × 4, so F = 10 N [1]. R = 20 N, so μ = 10 ÷ 20 = 0.5 [1].

Tip: write Newton’s second law for each particle separately, with the tension acting on both.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 42, Question 5.

5. (a) dv/dt = 6 − 2t = 0 when t = 3 [1]. v = 18 − 9 + 7 = 16 m s⁻¹ [1]. (b) s = ∫₀³ (6t − t² + 7) dt = [3t² − t³/3 + 7t]₀³ [1] = 27 − 9 + 21 = 39 m [1].

Tip: for maximum velocity, differentiate v and set the acceleration to zero.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 42, Question 6.

6. R = 50 cos 30° = 43.3 N [1]. The particle is about to slide down, so friction acts up the plane: F = 0.2 × 43.3 = 8.66 N [1]. Resolving along the plane: P + 8.66 = 50 sin 30° = 25 [1], so P = 16.3 N [1].

Tip: friction always opposes the motion that would happen, so decide which way the particle would slide.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 42, Question 5(b).

7. (a) Going up a distance d, the particle gains PE 2 × 10 × d sin 30° = 10d and does 5d of work against friction [1]. ½ × 2 × 6² = 36 = 10d + 5d [1], so d = 2.4 m [1]. (b) Coming down, PE lost = 10 × 2.4 = 24 J and work against friction = 5 × 2.4 = 12 J [1]. KE at the start = 24 − 12 = 12 J [1], so ½ × 2 × v² = 12 and v = 3.46 m s⁻¹ [1].

Tip: friction takes energy away in both directions, so the particle returns more slowly than it left.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 42, Question 7.

---

Where marks are usually lost

• The full force, rather than its component, used for work done.
• Velocities given the wrong sign in momentum questions.
• Friction drawn the wrong way on an inclined plane.
• Maximum velocity found by setting v = 0 instead of dv/dt = 0.