A-LevelMathematicsMechanics9709/42 · Oct/Nov 2021 · Q57 Marks
A railway engine of mass 75 000 kg is moving up a straight hill inclined at an angle α to the horizontal, where sin α = 0.01. The engine is travelling at a constant speed of 30 ms⁻¹. The engine is working at 960 kW. There is a constant force resisting the motion of the engine.
- (a)Find the resistance force.[3]
- (b)The engine comes to a section of track which is horizontal. At the start of the section the engine is travelling at 30 m s⁻¹ and the power of the engine is now reduced to 900 kW. The resistance to motion is no longer constant, but in the next 60 s the work done against the resistance force is 46 500 kJ. Find the speed of the engine at the end of the 60 s.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.
- B1 Allow for 960 000 = DF × 30. Driving force = DF = 960 000 / 30
- M1 Resolve forces along the slope. Must use a value for either sin α or α.
- A1 Allow correct work with 24500 to 3 sf. Resistance force = R = 24 500 N
- B1 WD by engine in 60 s = 900 000 × 60 [= 54000000]
- B1 For either correct expression for KE. KE_init = 0.5 × 75000 × 30² KE_final = 0.5 × 75000 × v²
- M1 For use of the work-energy equation with 4 terms, correct dimensions.
- A1 Allow v = √1100 = 10√11. Speed of engine after 60 s = v = 33.2 ms⁻¹
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