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CAIE M1 2015 June Q5
6 marks Standard +0.3
5 A particle \(P\) starts from rest at a point \(O\) on a horizontal straight line. \(P\) moves along the line with constant acceleration and reaches a point \(A\) on the line with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the instant that \(P\) leaves \(O\), a particle \(Q\) is projected vertically upwards from the point \(A\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Subsequently \(P\) and \(Q\) collide at \(A\). Find
  1. the acceleration of \(P\),
  2. the distance \(O A\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-3_538_414_315_370} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-3_561_686_264_1080} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Two particles \(P\) and \(Q\) have masses \(m \mathrm {~kg}\) and \(( 1 - m ) \mathrm { kg }\) respectively. The particles are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. \(P\) is held at rest with the string taut and both straight parts of the string vertical. \(P\) and \(Q\) are each at a height of \(h \mathrm {~m}\) above horizontal ground (see Fig. 1). \(P\) is released and \(Q\) moves downwards. Subsequently \(Q\) hits the ground and comes to rest. Fig. 2 shows the velocity-time graph for \(P\) while \(Q\) is moving downwards or is at rest on the ground.
CAIE M1 2015 June Q7
12 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-4_657_618_255_760} A small ring \(R\) is attached to one end of a light inextensible string of length 70 cm . A fixed rough vertical wire passes through the ring. The other end of the string is attached to a point \(A\) on the wire, vertically above \(R\). A horizontal force of magnitude 5.6 N is applied to the point \(J\) of the string 30 cm from \(A\) and 40 cm from \(R\). The system is in equilibrium with each of the parts \(A J\) and \(J R\) of the string taut and angle \(A J R\) equal to \(90 ^ { \circ }\) (see diagram).
  1. Find the tension in the part \(A J\) of the string, and find the tension in the part \(J R\) of the string. The ring \(R\) has mass 0.2 kg and is in limiting equilibrium, on the point of moving up the wire.
  2. Show that the coefficient of friction between \(R\) and the wire is 0.341 , correct to 3 significant figures. A particle of mass \(m \mathrm {~kg}\) is attached to \(R\) and \(R\) is now in limiting equilibrium, on the point of moving down the wire.
  3. Given that the coefficient of friction is unchanged, find the value of \(m\). \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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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 2015 June Q1
4 marks Easy -1.2
1 A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is 500 N and the block moves at a constant speed of \(2.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done by the tension in 40 s and find the power applied by the tension.
CAIE M1 2015 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{543cb1dd-40e8-4d66-8ca0-4f183f83f366-2_438_903_488_623} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a smooth horizontal surface with the string passing over a small smooth pulley fixed at the edge of the surface. \(B\) hangs vertically below the pulley at a distance \(h \mathrm {~m}\) above the floor (see diagram). \(A\) is released and the particles move. \(B\) reaches the floor and \(A\) subsequently reaches the pulley with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Explain briefly why the speed with which \(B\) reaches the floor is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(h\).
CAIE M1 2015 June Q3
6 marks Standard +0.3
3 A car of mass 860 kg travels along a straight horizontal road. The power provided by the car's engine is \(P\) W and the resistance to the car's motion is \(R \mathrm {~N}\). The car passes through one point with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The car passes through another point with speed \(22.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).
CAIE M1 2015 June Q4
6 marks Moderate -0.3
4 A lorry of mass 12000 kg moves up a straight hill of length 500 m , starting at the bottom with a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and reaching the top with a speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 25 m above the level of the bottom of the hill. The resistance to motion of the lorry is 7500 N . Find the driving force of the lorry. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{543cb1dd-40e8-4d66-8ca0-4f183f83f366-3_566_405_264_868} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Four coplanar forces of magnitudes \(4 \mathrm {~N} , 8 \mathrm {~N} , 12 \mathrm {~N}\) and 16 N act at a point. The directions in which the forces act are shown in Fig. 1.
  1. Find the magnitude and direction of the resultant of the four forces. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{543cb1dd-40e8-4d66-8ca0-4f183f83f366-3_351_629_1260_758} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The forces of magnitudes 4 N and 16 N exchange their directions and the forces of magnitudes 8 N and 12 N also exchange their directions (see Fig. 2).
  2. State the magnitude and direction of the resultant of the four forces in Fig. 2.
CAIE M1 2015 June Q6
9 marks Standard +0.3
6 A small box of mass 5 kg is pulled at a constant speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of a rough plane inclined at \(10 ^ { \circ }\) to the horizontal. The pulling force has magnitude 20 N and acts downwards parallel to a line of greatest slope of the plane.
  1. Find the coefficient of friction between the box and the plane. The pulling force is removed while the box is moving at \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the distance moved by the box after the instant at which the pulling force is removed.
    [0pt] [Question 7 is printed on the next page.]
CAIE M1 2015 June Q7
13 marks Moderate -0.3
7 A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O 100 \mathrm {~s}\) later. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\), where $$v = 0.0001 t ^ { 3 } - 0.015 t ^ { 2 } + 0.5 t$$
  1. Show that \(P\) is instantaneously at rest when \(t = 0 , t = 50\) and \(t = 100\).
  2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocitytime graph for \(P\) 's motion for \(0 \leqslant t \leqslant 100\).
  3. Find the greatest distance of \(P\) from \(O\) for \(0 \leqslant t \leqslant 100\). \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.
    To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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 2016 June Q1
5 marks Easy -1.2
1 A lift moves upwards from rest and accelerates at \(0.9 \mathrm {~ms} ^ { - 2 }\) for 3 s . The lift then travels for 6 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 4 s .
  1. Sketch a velocity-time graph for the motion.
  2. Find the total distance travelled by the lift.
CAIE M1 2016 June Q2
5 marks Moderate -0.8
2 A box of mass 25 kg is pulled, at a constant speed, a distance of 36 m up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves up a line of greatest slope against a constant frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope. Find
  1. the work done against friction,
  2. the change in gravitational potential energy of the box,
  3. the work done by the pulling force.
CAIE M1 2016 June Q3
6 marks Moderate -0.3
3 A car of mass 1000 kg is moving along a straight horizontal road against resistances of total magnitude 300 N .
  1. Find, in kW , the rate at which the engine of the car is working when the car has a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the acceleration of the car when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at \(90 \%\) of the power found in part (i).
CAIE M1 2016 June Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{099c81e0-a95a-4f98-801c-32d905ef7c7d-2_446_752_1521_699} Coplanar forces of magnitudes \(50 \mathrm {~N} , 48 \mathrm {~N} , 14 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Given that \(\tan \alpha = \frac { 7 } { 24 }\), find the values of \(P\) and \(\theta\).
CAIE M1 2016 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{099c81e0-a95a-4f98-801c-32d905ef7c7d-3_432_710_258_721} Two particles of masses 5 kg and 10 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The 5 kg particle is on a rough fixed slope which is at an angle of \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The 10 kg particle hangs below the pulley (see diagram). The coefficient of friction between the slope and the 5 kg particle is \(\frac { 1 } { 2 }\). The particles are released from rest. Find the acceleration of the particles and the tension in the string.
CAIE M1 2016 June Q6
9 marks Standard +0.3
6 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has a velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 6 t ^ { 2 } - 30 t + 24\).
  1. Find the set of values of \(t\) for which the acceleration of the particle is negative.
  2. Find the distance between the two positions at which \(P\) is at instantaneous rest.
  3. Find the two positive values of \(t\) at which \(P\) passes through \(O\).
CAIE M1 2016 June Q7
12 marks Standard +0.3
7 A particle of mass 30 kg is on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. Starting from rest, the particle is pulled up the plane by a force of magnitude 200 N acting parallel to a line of greatest slope.
  1. Given that the plane is smooth, find
    (a) the acceleration of the particle,
    (b) the change in kinetic energy after the particle has moved 12 m up the plane.
  2. It is given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.12 .
    (a) Find the acceleration of the particle.
    (b) The direction of the force of magnitude 200 N is changed, and the force now acts at an angle of \(10 ^ { \circ }\) above the line of greatest slope. Find the acceleration of the particle.
CAIE M1 2016 June Q1
5 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_373_591_260_776} Coplanar forces of magnitudes \(7 \mathrm {~N} , 6 \mathrm {~N}\) and 8 N act at a point in the directions shown in the diagram. Given that \(\sin \alpha = \frac { 3 } { 5 }\), find the magnitude and direction of the resultant of the three forces.
CAIE M1 2016 June Q2
5 marks Moderate -0.3
2 A particle \(P\) moves in a straight line, starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = 4 t ^ { 2 } - 8 t + 3\).
  1. Find the two values of \(t\) at which \(P\) is at instantaneous rest.
  2. Find the distance travelled by \(P\) between these two times.
CAIE M1 2016 June Q3
6 marks Standard +0.3
3 A particle of mass 8 kg is projected with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). The motion of the particle is resisted by a constant frictional force of magnitude 15 N . The particle comes to instantaneous rest after travelling a distance \(x \mathrm {~m}\) up the plane.
  1. Express the change in gravitational potential energy of the particle in terms of \(x\).
  2. Use an energy method to find \(x\).
CAIE M1 2016 June Q4
7 marks Moderate -0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_522_959_1692_593} A sprinter runs a race of 400 m . His total time for running the race is 52 s . The diagram shows the velocity-time graph for the motion of the sprinter. He starts from rest and accelerates uniformly to a speed of \(8.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 s . The sprinter maintains a speed of \(8.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 36 s , and he then decelerates uniformly to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race.
  1. Calculate the distance covered by the sprinter in the first 42 s of the race.
  2. Show that \(V = 7.84\).
  3. Calculate the deceleration of the sprinter in the last 10 s of the race.
CAIE M1 2016 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_394_531_260_806} A block of mass 2.5 kg is placed on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The block is kept in equilibrium by a light string making an angle of \(20 ^ { \circ }\) above a line of greatest slope. The tension in the string is \(T \mathrm {~N}\), as shown in the diagram. The coefficient of friction between the block and plane is \(\frac { 1 } { 4 }\). The block is in limiting equilibrium and is about to move up the plane. Find the value of \(T\).
CAIE M1 2016 June Q6
8 marks Moderate -0.8
6 A car of mass 1100 kg is moving on a road against a constant force of 1550 N resisting the motion.
  1. The car moves along a straight horizontal road at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (a) Calculate, in kW , the power developed by the engine of the car.
    (b) Given that this power is suddenly decreased by 22 kW , find the instantaneous deceleration of the car.
  2. The car now travels at constant speed up a straight road inclined at \(8 ^ { \circ }\) to the horizontal, with the engine working at 80 kW . Assuming the resistance force remains the same, find this constant speed.
CAIE M1 2016 June Q7
12 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_378_1001_1672_573} A particle \(A\) of mass 1.6 kg rests on a horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass 2.4 kg which hangs freely below the pulley. The system is released from rest with the string taut and with \(B\) at a height of 0.5 m above the ground, as shown in the diagram. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground.
  1. Given that the table is smooth, find the time taken by \(B\) to reach the ground.
  2. Given instead that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac { 3 } { 8 }\), find the total distance travelled by \(A\). You may assume that \(A\) does not reach the pulley.
CAIE M1 2016 June Q1
4 marks Moderate -0.3
1 A particle of mass 8 kg is pulled at a constant speed a distance of 20 m up a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal by a force acting along a line of greatest slope.
  1. Find the change in gravitational potential energy of the particle.
  2. The total work done against gravity and friction is 1146 J . Find the frictional force acting on the particle.
CAIE M1 2016 June Q2
5 marks Moderate -0.3
2 Alan starts walking from a point \(O\), at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along a horizontal path. Ben walks along the same path, also starting from \(O\). Ben starts from rest 5 s after Alan and accelerates at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . Ben then continues to walk at a constant speed until he is at the same point, \(P\), as Alan.
  1. Find how far Ben has travelled when he has been walking for 5 s and find his speed at this instant.
  2. Find the distance \(O P\).
CAIE M1 2016 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{aaf655c6-47f0-4f17-9a57-58aaf48728df-2_586_611_1171_767} The coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).