Three particles A, B and C of masses 5kg, 1kg and 2kg respectively lie at rest in that order on a straight smooth horizontal track XYZ. Initially A is at X, B is at Y and C is at Z. Particle A is projected towards B with a speed of 6ms⁻¹ and at the same instant C is projected towards B with a speed of vms⁻¹. In the subsequent motion, A collides and coalesces with B to form particle D. Particle D then collides and coalesces with C to form particle E and E moves towards Z.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [M1] 3 non-zero terms; allow sign errors; using correct masses. For reference VD = 5. If mgv used, allow M1 M1 A0 max.
2. [DM1] 6 non-zero terms; allow sign errors; using correct masses; allow their numerical VD. Allow v = VE for this mark. Note: 5×6-2v=(5+1+2)VE is M2. If mgv used, allow M1 M1 A0 max.
3. [A1] AG Must in terms of v, as v is given in the question or explicitly defined their letter used as v. Do not allow v=VE for this mark. Any error seen is A0 but condone saying 'divide by 2' or equivalent. If mgv used, allow M1 M1 A0 max.
4. [B1] For either KEinitial or KEfinal correct.
5. [M1] Using (sum of two initial KEs) – KEfinal = ±63. Correct number of relevant terms – correct masses, must be adding 2 KE terms for KEinitial (sum of two initial KEs) and KEfinal coming from use of correct formula and of the correct form.
6. [A1] AG Any error seen is A0. Allow solving correct quadratic expression, rather than correct quadratic equation, for full marks. If v = -13 seen it must be discarded. Must see solving for this mark. A quadratic equation followed by the answer is insufficient.
7. [B1]
8. [B1FT] FT their 6 which MUST come from 6t = 36.
9. [DM1] Using their VD from part (a). VD ≠ 6 or 3 and their80≠98.
10. [A1] Do not ISW.
11. [B1]