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Learn/ A Level Mathematics/ Past Papers/ 2021/ Oct/Nov 2021 /Paper 4 (Variant 3)

Mechanics — Variant 3

A Level Mathematics · 9709/43 · Oct/Nov 2021

Question Paper

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Mark Scheme

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Questions

5 of 6 questions shown here, checked against the paper. The others are in the PDF above — they contain diagrams or maths we could not yet verify, and we would rather show you the original than a transcription we are not sure of.

1
A metal post is driven vertically into the ground by dropping a heavy object onto it from above. The mass of the object is 120120 kg and the mass of the post is 4040 kg (see diagram). The object hits the post with speed 88 m s−1\text{m s}^{-1} and remains in contact with it after the impact.
1(a) [2]
Calculate the speed with which the combined post and object moves immediately after the impact.
1(b) [3]
There is a constant force resisting the motion of magnitude 48004800 N. Calculate the distance the post is driven into the ground.
2
A particle of mass 88 kg is suspended in equilibrium by two light inextensible strings which make angles of 60∘60^\circ and 45∘45^\circ above the horizontal.
2(a) [1]
Draw a diagram showing the forces acting on the particle.
2(b) [6]
Find the tensions in the strings.
3
A ball of mass 1.61.6 kg is released from rest at a point 55 m above horizontal ground. When the ball hits the ground it instantaneously loses 88 J of kinetic energy and starts to move upwards.
3(a) [3]
Use an energy method to find the greatest height that the ball reaches after hitting the ground.
3(b) [3]
Find the total time taken, from the initial release of the ball until it reaches this greatest height.
4
A car of mass 14001400 kg is moving on a straight road against a constant force of 12501250 N resisting the motion.
4(a)
The car moves along a horizontal section of the road at a constant speed of 3636 m s−1\text{m s}^{-1}.
4(a)(i) [2]
Calculate the work done against the resisting force during the first 88 seconds.
4(a)(ii) [2]
Calculate, in kW, the power developed by the engine of the car.
4(a)(iii) [3]
Given that this power is suddenly increased by 1212 kW, find the instantaneous acceleration of the car.
4(b) [2]
The car now travels at a constant speed of 3232 m s−1\text{m s}^{-1} up a section of the road inclined at θ∘\theta^\circ to the horizontal, with the engine working at 6464 kW. Find the value of θ\theta.
5
A particle PP moves in a straight line, starting from rest at a point OO on the line. At time tt s after leaving OO the acceleration of PP is k(16−t2)k(16 - t^{2}) m s−2\text{m s}^{-2}, where kk is a positive constant, and the displacement from OO is ss m. The velocity of PP is 88 m s−1\text{m s}^{-1} when t=4t = 4.
5(a) [5]
Show that s=164t2(96−t2)s = \frac{1}{64}t^{2}(96 - t^{2}).
5(b) [3]
Find the speed of PP at the instant that it returns to OO.
5(c) [3]
Find the maximum displacement of the particle from OO.

Published by Cambridge Assessment International Education. Reproduced for study use.