A-LevelMathematicsMechanics9709/43 · May/June 2025 · Q59 Marks

A van of mass 2500kg travelling at speed vms⁻¹ experiences a resistance force of kv² N. The constant power of the van's engine is 62.5 kW.

(a)The steady speed that the van could maintain when moving along a straight horizontal road is 50ms⁻¹. Show that k = 0.5, and find the acceleration of the van when its speed is 25ms⁻¹ on this straight horizontal road.[4]
(b)The van begins to ascend a hill inclined at an angle θ° to the horizontal. The van travels along a line of greatest slope of the hill. The speed of the van at the start of the hill is 20ms⁻¹, and its acceleration is 5ams⁻². Later, on the same hill, the speed of the van is 30ms⁻¹, and its acceleration is ams⁻². The power of the van's engine remains at 62.5kW, and the resistance force remains at 0.5v² N. Find the value of a and the value of θ.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[M1] For use of Power = DF×v - allow 62.5×103 for 62500, allow 62500=k×50³.
[A1] k = 0.5 AG-allow a correct equation followed by k=0.5.
[M1] For N2L with 3 terms; dimensionally correct but allow sign errors. If using 62500/50 (=1250) for the DF then M0.
[A1] Acceleration = 0.875 m s⁻² Allow 7/8
[*M1] With 4 relevant terms; allow sign errors; Allow sin/cos mix; condone 30 with 5a, 20 with a, but must be dimensionally correct.
[A2] A1 for either correct equation.
[DM1] For attempt to solve for a or θ – from equations with the correct number of relevant terms.
[A1] θ=3[.00] and a = 0.129 Allow a=31/240 , 0.129167... Allow 0.130 (0.129973...) from using θ=3 but not 0.13 unless greater accuracy seen.

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