A railway engine of mass 120000kg is towing a coach of mass 60000kg up a straight track inclined at an angle of α to the horizontal where sin α = 0.02. There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000N and there are constant resistances to motion of 22000 N on the engine and 13000N on the coach.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [M1] Engine: 125000-120000g×0.02-22000-T=120000a 125000-120000g sin(1.145.....) – 22000 – T = 120000a [125000-24000-22000-T =120000a ⇒ 79000-T =120000a] Attempt at Newton's second law at least once; correct number of relevant terms; allow sign errors; allow sin/cos mix; allow g missing; a value for a or sinα must be substituted. Allow with α=1.1 or better [α = 1.145991998].
2. [A1] Coach: T-60000g×0.02-13000=60000a T-60000g sin (1.145.....)–13000 = 60000a [T-12000-13000= 60000a ⇒ T-25000 = 60000a] Any equations correct.
3. [A1] System: 125000-120000g × 0.02 – 60000g × 0.02 – 22000-13000 = (120000+60000) a 125000-120000gsin (1.145.....) – 60000gsin (1.145.....) – 22000 – 13000 = (120000+60000) a [125000-24000-12000-22000-13000 = (120000+60000) a ⇒54000=180000a] Two equations correct. If using separate equations for engine and coach and different T's, then allow M1A1A0 max.
4. [DM1] Solve for T or a Using equations with the correct number of relevant terms. If no working seen, must be solutions to their equation(s) to be awarded M1.
5. [A1] Acceleration = 0.3 ms⁻² and Tension = 43000 N Allow 0.299 from use of α=1.15. Awrt 43000 to 3sf from correct work.
6. [B1] Driving force, DF = 4500000 / 30 [=150000] Use of F = P/v oe e.g. DF×30=4500000.
7. [M1] System: 150000-120000gsinβ – 60000gsinβ – 22000-13000=0 or for Engine: 150000-120000gsinβ – 22000 – T'=0 and Coach: T'-60000gsinβ–13000=0 Attempt to resolve parallel to the track once if using system equation, twice if using equations for engine and coach separately Correct number of relevant terms; allow sign errors; allow sin/cos mix; allow g missing. Must be correct number of equations depending on method.
8. [A1] Allow DF or their DF. Must be using same T'.
9. [A1] Solve to get β=3.7° 3.663058552 awrt 3.7° www.