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CAIE M1 2010 June Q3
5 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-2_241_511_1676_817} A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at \(45 ^ { \circ }\) to the horizontal (see diagram).
  1. Show that the normal component of the contact force acting on the ring has magnitude 3.05 N , correct to 3 significant figures.
  2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod.
CAIE M1 2010 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-3_755_561_248_790} Coplanar forces of magnitudes \(250 \mathrm {~N} , 160 \mathrm {~N}\) and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. \(5 P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of 0.45 m above the level of \(P\). A particle of mass 0.3 kg moves upwards along the line \(P Q\).
  1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\).
  2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is 0.39 J . Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\).
CAIE M1 2010 June Q6
11 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-4_451_729_255_708} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m . The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.
  1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley.
CAIE M1 2010 June Q7
11 marks Standard +0.3
7 A vehicle is moving in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after the vehicle starts is given by $$\begin{aligned} & v = A \left( t - 0.05 t ^ { 2 } \right) \quad \text { for } 0 \leqslant t \leqslant 15 , \\ & v = \frac { B } { t ^ { 2 } } \quad \text { for } t \geqslant 15 , \end{aligned}$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is 225 m .
  1. Find the value of \(A\) and show that \(B = 3375\).
  2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geqslant 15\).
  3. Find the speed of the vehicle when it has travelled a total distance of 315 m . \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2010 June Q2
5 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-2_661_1351_479_397} The diagram shows the velocity-time graph for the motion of a machine's cutting tool. The graph consists of five straight line segments. The tool moves forward for 8 s while cutting and then takes 3 s to return to its starting position. Find
  1. the acceleration of the tool during the first 2 s of the motion,
  2. the distance the tool moves forward while cutting,
  3. the greatest speed of the tool during the return to its starting position.
CAIE M1 2010 June Q3
5 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-2_241_511_1676_817} A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at \(45 ^ { \circ }\) to the horizontal (see diagram).
  1. Show that the normal component of the contact force acting on the ring has magnitude 3.05 N , correct to 3 significant figures.
  2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod.
CAIE M1 2010 June Q4
7 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-3_755_561_248_790} Coplanar forces of magnitudes \(250 \mathrm {~N} , 160 \mathrm {~N}\) and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. \(5 P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of 0.45 m above the level of \(P\). A particle of mass 0.3 kg moves upwards along the line \(P Q\).
  1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\).
  2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is 0.39 J . Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\).
CAIE M1 2010 June Q6
11 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-4_451_729_255_708} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m . The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.
  1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley.
CAIE M1 2010 June Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-2_582_751_255_696} Three coplanar forces act at a point. The magnitudes of the forces are \(5.5 \mathrm {~N} , 6.8 \mathrm {~N}\) and 7.3 N , and the directions in which the forces act are as shown in the diagram. Given that the resultant of the three forces is in the same direction as the force of magnitude 6.8 N , find the value of \(\alpha\) and the magnitude of the resultant.
CAIE M1 2010 June Q2
5 marks Moderate -0.3
2 A particle starts at a point \(O\) and moves along a straight line. Its velocity \(t\) s after leaving \(O\) is \(\left( 1.2 t - 0.12 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the displacement of the particle from \(O\) when its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2010 June Q3
7 marks Moderate -0.3
3 A load is pulled along a horizontal straight track, from \(A\) to \(B\), by a force of magnitude \(P \mathrm {~N}\) which acts at an angle of \(30 ^ { \circ }\) upwards from the horizontal. The distance \(A B\) is 80 m . The speed of the load is constant and equal to \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it moves from \(A\) to the mid-point \(M\) of \(A B\).
  1. For the motion from \(A\) to \(M\) the value of \(P\) is 25 . Calculate the work done by the force as the load moves from \(A\) to \(M\). The speed of the load increases from \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it moves from \(M\) towards \(B\). For the motion from \(M\) to \(B\) the value of \(P\) is 50 and the work done against resistance is the same as that for the motion from \(A\) to \(M\). The mass of the load is 35 kg .
  2. Find the gain in kinetic energy of the load as it moves from \(M\) to \(B\) and hence find the speed with which it reaches \(B\).
CAIE M1 2010 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-3_499_567_260_788} The diagram shows a vertical cross-section of a triangular prism which is fixed so that two of its faces are inclined at \(60 ^ { \circ }\) to the horizontal. One of these faces is smooth and one is rough. Particles \(A\) and \(B\), of masses 0.36 kg and 0.24 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the highest point of the cross-section. \(B\) is held at rest at a point of the cross-section on the rough face and \(A\) hangs freely in contact with the smooth face (see diagram). \(B\) is released and starts to move up the face with acceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. By considering the motion of \(A\), show that the tension in the string is 3.03 N , correct to 3 significant figures.
  2. Find the coefficient of friction between \(B\) and the rough face, correct to 2 significant figures.
CAIE M1 2010 June Q5
8 marks Moderate -0.3
5 A ball moves on the horizontal surface of a billiards table with deceleration of constant magnitude \(d \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The ball starts at \(A\) with speed \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and reaches the edge of the table at \(B , 1.2 \mathrm {~s}\) later, with speed \(1.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the distance \(A B\) and the value of \(d\). \(A B\) is at right angles to the edge of the table containing \(B\). The table has a low wall along each of its edges and the ball rebounds from the wall at \(B\) and moves directly towards \(A\). The ball comes to rest at \(C\) where the distance \(B C\) is 2 m .
  2. Find the speed with which the ball starts to move towards \(A\) and the time taken for the ball to travel from \(B\) to \(C\).
  3. Sketch a velocity-time graph for the motion of the ball, from the time the ball leaves \(A\) until it comes to rest at \(C\), showing on the axes the values of the velocity and the time when the ball is at \(A\), at \(B\) and at \(C\).
CAIE M1 2010 June Q6
9 marks Standard +0.3
6 Particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. \(P\) is released from rest at a point \(O\) on the line and 2 s later passes through the point \(A\) with speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the acceleration of \(P\) and the angle of inclination of the plane. At the instant that \(P\) passes through \(A\) the particle \(Q\) is released from rest at \(O\). At time \(t\) s after \(Q\) is released from \(O\), the particles \(P\) and \(Q\) are 4.9 m apart.
  2. Find the value of \(t\).
CAIE M1 2010 June Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-4_246_665_253_739} Two rectangular boxes \(A\) and \(B\) are of identical size. The boxes are at rest on a rough horizontal floor with \(A\) on top of \(B\). Box \(A\) has mass 200 kg and box \(B\) has mass 250 kg . A horizontal force of magnitude \(P\) N is applied to \(B\) (see diagram). The boxes remain at rest if \(P \leqslant 3150\) and start to move if \(P > 3150\).
  1. Find the coefficient of friction between \(B\) and the floor. The coefficient of friction between the two boxes is 0.2 . Given that \(P > 3150\) and that no sliding takes place between the boxes,
  2. show that the acceleration of the boxes is not greater than \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
  3. find the maximum possible value of \(P\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2011 June Q1
4 marks Moderate -0.8
1 A car of mass 700 kg is travelling along a straight horizontal road. The resistance to motion is constant and equal to 600 N .
  1. Find the driving force of the car's engine at an instant when the acceleration is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that the car's speed at this instant is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the rate at which the car's engine is working.
CAIE M1 2011 June Q2
5 marks Easy -1.2
2 A load of mass 1250 kg is raised by a crane from rest on horizontal ground, to rest at a height of 1.54 m above the ground. The work done against the resistance to motion is 5750 J .
  1. Find the work done by the crane.
  2. Assuming the power output of the crane is constant and equal to 1.25 kW , find the time taken to raise the load.
CAIE M1 2011 June Q3
6 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{d5acfe31-8614-4508-ac5b-865e15a1f539-2_661_565_1069_790} A small smooth ring \(R\) of weight 8.5 N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\), with \(A\) vertically above \(B\). A horizontal force of magnitude 15.5 N acts on \(R\) so that the ring is in equilibrium with angle \(A R B = 90 ^ { \circ }\). The part \(A R\) of the string makes an angle \(\theta\) with the horizontal and the part \(B R\) makes an angle \(\theta\) with the vertical (see diagram). The tension in the string is \(T \mathrm {~N}\). Show that \(T \sin \theta = 12\) and \(T \cos \theta = 3.5\) and hence find \(\theta\).
CAIE M1 2011 June Q4
7 marks Standard +0.3
4 A block of mass 11 kg is at rest on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. A force acts on the block in a direction up the plane parallel to a line of greatest slope. When the magnitude of the force is \(2 X \mathrm {~N}\) the block is on the point of sliding down the plane, and when the magnitude of the force is \(9 X \mathrm {~N}\) the block is on the point of sliding up the plane. Find
  1. the value of \(X\),
  2. the coefficient of friction between the block and the plane.
CAIE M1 2011 June Q5
8 marks Moderate -0.8
5 A train starts from rest at a station \(A\) and travels in a straight line to station \(B\), where it comes to rest. The train moves with constant acceleration \(0.025 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 600 s , with constant speed for the next 2600 s , and finally with constant deceleration \(0.0375 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the total time taken for the train to travel from \(A\) to \(B\).
  2. Sketch the velocity-time graph for the journey and find the distance \(A B\).
  3. The speed of the train \(t\) seconds after leaving \(A\) is \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). State the possible values of \(t\).
CAIE M1 2011 June Q6
9 marks Standard +0.3
6 A particle travels in a straight line from a point \(P\) to a point \(Q\). Its velocity \(t\) seconds after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t - \frac { 1 } { 16 } t ^ { 3 }\). The distance \(P Q\) is 64 m .
  1. Find the time taken for the particle to travel from \(P\) to \(Q\).
  2. Find the set of values of \(t\) for which the acceleration of the particle is positive.
CAIE M1 2011 June Q7
11 marks Standard +0.3
7 Loads \(A\) and \(B\), of masses 1.2 kg and 2.0 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical. \(A\) is released and starts to move upwards. It does not reach the pulley in the subsequent motion.
  1. Find the acceleration of \(A\) and the tension in the string.
  2. Find, for the first 1.5 metres of \(A\) 's motion,
    (a) A's gain in potential energy,
    (b) the work done on \(A\) by the tension in the string,
    (c) A's gain in kinetic energy. B hits the floor 1.6 seconds after \(A\) is released. \(B\) comes to rest without rebounding and the string becomes slack.
  3. Find the time from the instant the string becomes slack until it becomes taut again.
CAIE M1 2011 June Q1
4 marks Easy -1.3
1 A load is pulled along horizontal ground for a distance of 76 m , using a rope. The rope is inclined at \(5 ^ { \circ }\) above the horizontal and the tension in the rope is 65 N .
  1. Find the work done by the tension. At an instant during the motion the velocity of the load is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the rate of working of the tension at this instant.
CAIE M1 2011 June Q2
4 marks Moderate -0.3
2 An object of mass 8 kg slides down a line of greatest slope of an inclined plane. Its initial speed at the top of the plane is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its speed at the bottom of the plane is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against the resistance to motion of the object is 120 J . Find the height of the top of the plane above the level of the bottom.
CAIE M1 2011 June Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-2_748_1410_979_370} The velocity-time graph shown models the motion of a parachutist falling vertically. There are four stages in the motion:
  • falling freely with the parachute closed,
  • decelerating at a constant rate with the parachute open,
  • falling with constant speed with the parachute open,
  • coming to rest instantaneously on hitting the ground.
    1. Show that the total distance fallen is 1048 m .
The weight of the parachutist is 850 N .
  • Find the upward force on the parachutist due to the parachute, during the second stage.
    •