# CIE Oct 2021 9709 Mechanics Paper 42

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This page covers Questions and Worked Solutions for CIE Mechanics Paper 42 October/November 2021, 9709/42.

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1. The diagram shows a velocity-time graph which models the motion of a car. The graph consists of six straight line segments. The car accelerates from rest to a speed of 20 m s−1 over a period of 5 s, and then travels at this speed for a further 20 s. The car then decelerates to a speed of 6 m s−1 over a period of 5 s. This speed is maintained for a further (T − 30)s. The car then accelerates again to a speed of 20 m s−1 over a period of (50 − T)s, before decelerating to rest over a period of 10 s.
(a) Given that during the two stages of the motion when the car is accelerating, the accelerations are equal, find the value of T.
(b) Find the total distance travelled by the car during the motion.
2. A van of mass 3600 kg is towing a trailer of mass 1200 kg along a straight horizontal road using a light horizontal rope. There are resistance forces of 700 N on the van and 300 N on the trailer.
(a) The driving force exerted by the van is 2500 N.
Find the tension in the rope.
The driving force is now removed and the van driver applies a braking force which acts only on the van. The resistance forces remain unchanged.
(b) Find the least possible value of the braking force which will cause the rope to become slack.
3. The diagram shows a semi-circular track ABC of radius 1.8 m which is fixed in a vertical plane. The points A and C are at the same horizontal level and the point B is at the bottom of the track. The section AB is smooth and the section BC is rough. A small block is released from rest at A.
(a) Show that the speed of the block at B is 6 m s−1.
The block comes to instantaneous rest for the first time at a height of 1.2 m above the level of B. The work done against the resistance force during the motion of the block from B to this point is 4.5 J.
(b) Find the mass of the block.
4. A cyclist starts from rest at a point A and travels along a straight road AB, coming to rest at B. The displacement of the cyclist from A at time ts after the start is s m, where s = 0.004(75t2 − t3).
(a) Show that the distance AB is 250 m.
(b) Find the maximum velocity of the cyclist.

1. A railway engine of mass 75 000 kg is moving up a straight hill inclined at an angle ! to the horizontal, where sin α = 0.01. The engine is travelling at a constant speed of 30 m s−1. The engine is working at 960 kW. There is a constant force resisting the motion of the engine.
(a) Find the resistance force.
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.
(b) Find the speed of the engine at the end of the 60 s.
2. A block of mass 5 kg is held in equilibrium near a vertical wall by two light strings and a horizontal force of magnitude X N, as shown in the diagram. The two strings are both inclined at 60° to the vertical.
(a) Given that X = 100, find the tension in the lower string.
(b) Find the least value of X for which the block remains in equilibrium in the position shown.
3. Particles P and Q have masses m kg and 2m kg respectively. The particles are initially held at rest 6.4 m apart on the same line of greatest slope of a rough plane inclined at an angle α to the horizontal, where sin α = 0.8 (see diagram). Particle P is released from rest and slides down the line of greatest slope. Simultaneously, particle Q is projected up the same line of greatest slope at a speed of 10m s−1.
The coefficient of friction between each particle and the plane is 0.6.
(a) Show that the acceleration of Q up the plane is −11.6 m s−2.
(b) Find the time for which the particles are in motion before they collide.
(c) The particles coalesce on impact.
Find the speed of the combined particle immediately after the impact.
4. Two particles A and B, of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to a horizontal plane and to the top of an inclined plane. The particles are initially at rest with A on the horizontal plane and B on the inclined plane, which makes an angle of 30° with the horizontal. The string is taut and B can move on a line of greatest slope of the inclined plane. A force of magnitude 3.5 N is applied to B acting down the plane (see diagram).
(a) Given that both planes are smooth, find the tension in the string and the acceleration of B.
(b) It is given instead that the two planes are rough. When each particle has moved a distance of 0.6 m from rest, the total amount of work done against friction is 1.1 J.
Use an energy method to find the speed of B when it has moved this distance down the plane.
[You should assume that the string is sufficiently long so that A does not hit the pulley when it moves 0.6 m.]

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