A-level mechanics questions, worked through step by step

What the formula booklet gives you

The constant acceleration (SUVAT) equations are in AQA's booklet, in both scalar and vector form. Edexcel's specification says you must know F = ma, and v = dr/dt and a = dv/dt for variable acceleration (Appendix 1). The skill the booklet can't give you is drawing a good force diagram.

Kinematics: SUVAT questions

Question 1: a cyclist sets off from rest with a constant acceleration of 0.8 m s⁻², which lasts for 15 s. Find the final speed and the distance covered.

List u = 0, a = 0.8, t = 15. Then v = u + at = 12 m s⁻¹, and s = ut + ½at² = ½ × 0.8 × 225 = 90 m.

Question 2: a stone is dropped from rest from a height of 45 m. How long does it take to reach the ground? Taking down as positive, u = 0, a = 9.8 and s = 45, so 45 = ½ × 9.8 × t², giving t = √(90 ÷ 9.8) = 3.0 s to 2 significant figures.

Choose a positive direction and stick to it. The SUVAT equations guide has all five equations, and the SUVAT Calculator checks answers. Can you pick the right equation every time? Try this from our Laws of Motion course.

Forces: connected particles

Question 3: particles of mass 5 kg and 3 kg hang either side of a smooth fixed pulley on a light inextensible string. Find the acceleration and the tension.

1. For the 5 kg particle, moving down: 5g − T = 5a.
2. For the 3 kg particle, moving up: T − 3g = 3a.
3. Add them: 2g = 8a, so a = 2.45 m s⁻².
4. Substitute: T = 3(9.8 + 2.45) = 36.75 N, which is 36.8 N to 3 significant figures.

"Light", "smooth" and "inextensible" are modelling words: they mean the tension is the same throughout and both particles share one acceleration.

Forces: friction on a slope

Question 4: a 2 kg particle rests on a rough plane inclined at 30° to the horizontal. Find the normal reaction, the friction force and the least possible coefficient of friction.

Resolve perpendicular to the plane: R = 2g cos 30° = 17.0 N. Resolve along the plane: F = 2g sin 30° = 9.8 N. For it to stay at rest, F ≤ μR, so μ ≥ 9.8 ÷ 17.0 = 0.577, which is tan 30°. Resolve along and perpendicular to the slope, not horizontally and vertically, and the algebra stays short.

Moments

Question 5: a uniform beam AB, 4 m long with mass 20 kg, rests on supports at A and B. A 30 kg child stands 1 m from A. Find the reaction at each support.

Being uniform, the beam has its weight at the midpoint, 2 m from A. Take moments about A to remove the reaction there: 4R_B = 20g × 2 + 30g × 1 = 70g, so R_B = 17.5g = 171.5 N. Vertically, R_A + R_B = 50g, so R_A = 32.5g = 318.5 N. The support nearer the child takes more weight, a quick check that the answer makes sense. Try more with the Moments Calculator.

Kinematics with calculus

Question 6: a particle's velocity is v = 3t² − 2t m s⁻¹. Find its acceleration when t = 2, and its displacement from t = 0 to t = 2.

Differentiate for acceleration: a = 6t − 2, so a = 10 m s⁻² at t = 2. Integrate for displacement: ∫₀² (3t² − 2t) dt = [t³ − t²] from 0 to 2 = 8 − 4 = 4 m. SUVAT can't be used here, because the acceleration changes with time.

Common A-level mechanics mistakes

• SUVAT with changing acceleration. Use calculus whenever a depends on t.
• No force diagram. Draw one for every forces question, with every force labelled.
• Signs. Pick a positive direction and keep to it.
• Weight as m instead of mg. Weight is a force, in newtons.
• Moments about the wrong point. Take moments about a point where an unknown force acts, to remove it.
• Too many significant figures. Because g = 9.8 is given to two significant figures, round final answers to two or three.

Use a daily paper as a short timed mock

For more A-level mechanics questions, sit today's A-level Maths: Mechanics paper. For exam dates, papers and the formula booklet, see the A-level Maths guide.

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