Particles X and Y move in a straight line through points A and B. Particle X starts from rest at A and moves towards B. At the same instant, Y starts from rest at B. At time t seconds after the particles start moving • the acceleration of X in the direction AB is given by (12t + 12)ms-2, • the acceleration of Y in the direction AB is given by (24t – 8) ms-2.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [M1] v = ∫12t+12 dt, vx = 6t² +12t or v = ∫24t-8 dt, vy =12t² - 8t Uses v = ∫adt.
2. [A1] vx = 6t² +12t vy = 12t² - 8t All correct.
3. [B1] 6t² +12t=12t² -8t leading to t = 10/3 Solves vx = Vy .
4. [*M1] sx = ∫6t² +12tdt sx = 2t³ + 6t² Sy = ∫12t² - 8t dt sy = 4t³ – 4t² Uses s = ∫ vdt.
5. [DM1] Sx = 2(10/3)³ + 6(10/3)² Sy = 4(10/3)³ - 4(10/3)² Evaluates s(10/3) for each particle.
6. [A1] AB = 3800/27 - 2800/27 = 1000/27 Distance AB = 37 m
7. [M1] AB = −2t³ +10t² = −2×3³ +10×3² Calculates distance AB.
8. [A1] [t=3], AB = 36 m AG