A particle P of mass 3 kg is projected with a speed of 8 ms⁻¹ up a line of greatest slope of a rough plane inclined at 30° to the horizontal. P is projected from a point A on the plane and comes to instantaneous rest at a point B on the plane. P then slides back down the plane. The coefficient of friction between P and the plane is ¹⁄₁₂√3.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [1] Apply the work-energy principle for the motion from A to B to form an equation in one variable only Correct number of relevant terms, allow sign errors and sin/cos mix. Dimensionally correct, terms that need a component should have a component.
2. [1] 1/2×3×8²–3gd sin 30 A1 for correct LHS.
3. [1] -1/2 × 3g cos 30 × d (where d is the distance AB) [96-15d=3.75d] A1 for correct RHS.
4. [1] For either d = 5.12 m or work done against friction is 19.2 J Either value stated or clearly implied by later working. If using N2L then M0, but SCB1 only (see below).
5. [1] 1/2×3×v² = 1/2×3×8²–2x(their19.2) [1.5v² =96-38.4] Either consider the total work done against friction from A to B and B to A or, consider motion from B to A. In both cases must have the correct number of terms but allow sign errors and cos/sin mix. Dimensionally correct, terms that need a component should have a component. Dependent on previous M1 or previous SCB1.
6. [1] v = 6.20 m s⁻¹ ONLY 8√15/5, 6.196773354... must be positive. If using N2L then M0, but SCB1 only (see below). Allow 6.2 from CWO. A 3sf answer of 6.19 becoming 6.2 is A0. This mark is dependent on all previous 5 marks awarded or on the SCB1 and the previous M1.