A car of mass 1800kg is towing a trailer of mass 300kg up a straight road inclined at an angle α to the horizontal, where sinα = 0.05. The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of 800N acting on the car and a resistance force of FN acting on the trailer. The driving force of the car's engine is 3000N.

Mark scheme

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

1. [*M1] Attempt at Newton's second law on car or trailer or system. Correct number of terms; allow sign errors; allow sin/cos mix; sinα or α=2.865... needs to be substituted; allow with α=2.87or2.9 or better; allow g missing.
2. [A1] Car: 3000-1800g×0.05-800-T=1800a [3000-900-800-T=1800a -> 1300-T=1800a]. Trailer: T-300g×0.05-100=300a [T-150-100 = 300a -> T-250 = 300a]. System: 3000-1800g×0.05-300g×0.05-800-100 = (1800+300)a [3000-900-150-800-100 = 2100a -> 1050 = 2100a]
3. [DM1] Solving for a or T. From equations with correct number of relevant terms, allow g missing. If no working seen, must be correct for their equations.
4. [A1] Acceleration = 0.5 ms⁻². Oe. Allow 0.499 from α=2.87.
5. [A1] Tension = 400 N. Allow awrt 400 to 3sf, www. Condone using car equation with 0.499 to get T=400.5. Using exact α from an inexact angle gives T=400.
6. [B1] Work done against resistance on car = 800×50[=40000] or Work done by driving force = 3000×50[=150000]
7. [B1] PE change = (1800+300)g×50×0.05[= 52500]. Allow 2100g×50×sin2.9. Or 2100g×50×sin 2.87.
8. [B1] Initial KE = ½×(1800+300)×20² [420000]
9. [M1] Attempt at work-energy equation; correct number of relevant terms; dimensionally correct; allow sign errors; allow sin/cos mix in relevant resolved terms. Only PE must be from a component. Allow 2100g×50× sin 2.9 or 2100g×50× sin 2.87.
10. [A1] ½×(1800+300)v² = 3000×50-800×50-6000-(1800+300)g×50×0.05+½×(1800+300)×20² [1050v² = 150000-40000-6000-52500+420000] [1050v² = 471500]. Speed = 21.2ms⁻¹ [21.1907...]. Awrt 21.2 to 3sf. α=2.87 gives 21.18909187. α = 2.9 gives 21.17674855.