A particle P of mass 0.2 kg is released from rest at a point O on a rough plane inclined at 60° to the horizontal, and travels down a line of greatest slope. The coefficient of friction between P and the plane is 0.3. A force of magnitude 0.6x N acts on P in the direction PO, where x m is the displacement of P from O.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

1. [M1A1] 0.2vdv/dx=0.2gsin60 – 0.3 × 0.2gcos60 – 0.6x Uses Newton's Second Law parallel to the plane. Correct equation.
2. [A1] vdv/dx = 5√3 – 1.5 – 3x AG
3. [B1] x = (5√3 – 1.5)/3 (= 2.39) Uses a = 0.
4. [M1] ∫v dv = ∫(5√3 – 1.5 – 3x) dx Separates the variables and attempts to integrate.
5. [A1] v² /2 = 5 √3 x −1.5x − 3 x² /2 ( + c) Allow c = 0 without calculation seen.
6. [A1] v = 4.13 Substitutes x = 2.39.
7. [M1] 0 = 5 √3 x − 1.5x − 3 x² /2 Puts v = 0 and attempts to solve a quadratic equation.
8. [A1] x = 4.77(35...)
9. [M1] a = 5√3 - 1.5-3 × 4.77(35...)
10. [A1] Magnitude of a = 7.16 m s⁻²