Three particles A, B and C of masses 0.3 kg, 0.4kg and mkg respectively lie at rest in a straight line on a smooth horizontal plane. The distance between B and C is 2.1 m. A is projected directly towards B with speed 2 ms⁻¹. After A collides with B the speed of A is reduced to 0.6ms⁻¹, still moving in the same direction.

Mark scheme

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

1. [M1] For use of conservation of momentum in the A-B collision. Must be 3 terms. Allow sign errors. 0.3×2[+0]=0.3×0.6+0.4×v_B.
2. [A1] Speed of B = 1.05 ms⁻¹ (AG). Allow M1 A0 if g included with the masses.
3. [M1] For use of conservation of momentum in the B-C collision. Must be 3 terms. Allow sign errors. 0.4×1.05[+0]=(0.4+m)×0.5.
4. [A1] m = 0.44 kg or ¹¹⁄₂₅ kg. Allow M1 A0 if g included with the masses.
5. [B1] Calculate the time for B to reach C: t_BC = 2.1 m / 1.05 ms⁻¹ = 2 s. Calculate the distance A travels during this time: d_A = 0.6 ms⁻¹ × 2 s = 1.2 m. Or, state the initial separation distance between A and the point of B-C collision: 2.1 m - 1.2 m = 0.9 m.
6. [B1] Express the distance travelled by particle A relative to the point where B and C collide as a function of time t after the B-C collision. e.g., 0.6t or (initial separation + 0.6t). Assume constant velocity (a=0).
7. [B1] Express the distance travelled by the combined BC particle relative to its collision point as a function of time t. e.g., 0.5t. Assume constant velocity (a=0).
8. [M1] Formulate an equation for the collision to occur (e.g., equate positions or use relative distance/speed). e.g., 0.9 + 0.5t = 0.6t OR t = 0.9 / (0.6 - 0.5). Allow use of their initial separation and speeds. Do not allow M1 if acceleration terms are included without justification of a=0.
9. [A1] Time = 9 s (Correct Working Only).