A van of mass 4500 kg is towing a trailer of mass 750kg down a straight hill inclined at an angle of θ to the horizontal where sinθ = 0.05 . The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500N on the van and 300 N on the trailer.

Mark scheme

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

1. [M1*] Use of Newton's second law for van or trailer or system. Must have correct number of relevant terms. Allow sign errors. Allow sin/cos mix. Allow g missing. Masses must be correct for their equation(s). Forces must have components (or not) as required. Must have either 0.05 or sin2.86... or sin 2.9, not just sin θ.
2. [A1] Trailer: 450+750g × 0.05-300=750a [525 = 750a] OR Van: D+4500g × 0.05 – 2500-450 = 4500a [D-700 = 4500a]
3. [DM1] For attempt to solve for a or D. Must get to ‘a =' or 'D='. Must have correct number of relevant terms in the equation(s) which they are using to find a or D. g must be present. Allow sign errors. Allow sin/cos mix. If no working shown to solve their equations, then their answers should be correct for their equations.
4. [A1] a=0.7 ms-2 and D = 3850 N
5. [M1*] Use of Newton's second law for van or trailer or system. Must have correct number of relevant terms Allow sign errors. Allow sin/cos mix. Allow g missing. Masses must be appropriate for their equation(s). Forces must have components (or not) as required. Must have either 0.09 or sin 5.16... or sin 5.2 not just sin θ.
6. [A1A1] Trailer: T-300-750g×0.09 = 750a [T-975=750a] Van: 9100-2500 – 4500g × 0.09 – T = 4500a [2550-T = 4500a] A1 for one correct equation, second A1 for another correct equation. If using Van and Trailer equations, must be using the same T for both to get the second A1.
7. [DM1] For attempt to solve for a or T. Must get to ‘a =' or ‘T='. Must have correct number of relevant terms in the equation(s) which they are using to find a or T. g must be present. Allow sign errors. Allow sin/cos mix. If no working shown to solve their equations, then their answers should be correct for their equations.
8. [A1] T=1200 N and a 0.3 ms-2
9. [M1] For use of `v² = 20² + 2a×375` or other complete method to find v² or v.
10. [A1FT] v = 25 ms⁻¹ [FT their value of a, i.e. `v = √400 + 750×their a`. Provided it does not lead to root of negative value.]