A particle P moves in a straight line starting from a point O and comes to rest 35 s later. At time t s after leaving O, the velocity v m s⁻¹ of P is given by v = (4/5)t² 0 ≤ t ≤ 5, v = 2t + 10 5 ≤ t ≤ 15, v = a + bt² 15 ≤ t ≤ 35, where a and b are constants such that a > 0 and b < 0.

Mark scheme

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

1. [M1] For matching velocities at t = 15 and using v = 0 at t = 35
2. [M1] Solve for a and b
3. [A1] a = 49 and b = -0.04 AG
4. [B1] Increasing quadratic, from (0,0) to (5,20), concave up
5. [B1] Line from (5,20) to (15,40)
6. [B1] Decreasing quadratic, from (15,40) to (35,0), concave down
7. [B1] 20 and 40 seen correct on v-axis
8. [B1] A₁ = ∫(0 to 5) (4/5)t² dt = 100/3
9. [M1] Using trapezium rule or integration for t = 5 to t = 15
10. [M1] Attempt to integrate the quadratic function from t = 15 to t = 35
11. [A1] A₃ = 453.3333 = 1360/3
12. [A1] Total Distance = 2360/3 = 787 m