A particle P is projected horizontally from a point O on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.2. A horizontal force of magnitude 0.06t N directed away from O acts on P, where ts is the time after projection. P comes to rest when t = 4.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2019). The mark or mark type is shown in brackets.

1. [M1] Resolve along the plane
2. 0.2mg = 0.06 × 8
3. [AG]
4. m = 0.24 kg
5. [M1] Use N2L along the plane
6. m dv/dt = 0.06t - 0.2mg or 0.24 dv/dt = 0.06t - 0.2 × 0.24g
7. [AG]
8. dv/dt = 0.25t-2
9. [A1]
10. [M1] Attempt to integrate
11. ∫dv = ∫(0.25t - 2)dt
12. [M1] Attempt to find c
13. v = 0.25t² / 2 – 2t + c, Put v = 0 and t = 4 ( leads to c = 6 )
14. [A1]
15. Initial velocity = 6 m s¯¹
16. [M1] Attempt to integrate
17. x = ∫(0.25 t² /2 – 2t + 6)dt
18. [A1ft] ft candidates c from part (ii)
19. x = 0.25 t³ /6 – t² + 6t (+ k )
20. [M1]
21. Finds or assumes k = 0 and substitutes t = 4 OR uses limits of 0 and 4
22. [A1]
23. OP= 32/3 = 10 2/3 = 10.7 m