Questions M1 (1912 questions)

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CAIE M1 2014 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-4_342_1257_255_445} A smooth inclined plane of length 160 cm is fixed with one end at a height of 40 cm above the other end, which is on horizontal ground. Particles \(P\) and \(Q\), of masses 0.76 kg and 0.49 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle \(P\) is held at rest on the same line of greatest slope as the pulley and \(Q\) hangs vertically below the pulley at a height of 30 cm above the ground (see diagram). \(P\) is released from rest. It starts to move up the plane and does not reach the pulley. Find
  1. the acceleration of the particles and the tension in the string before \(Q\) reaches the ground,
  2. the speed with which \(Q\) reaches the ground,
  3. the total distance travelled by \(P\) before it comes to instantaneous rest.
CAIE M1 2015 June Q1
1 A block \(B\) of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(\theta\) above the horizontal. The normal component of the contact force acting on \(B\) has magnitude 20 N .
  1. Show that \(\sin \theta = 0.28\).
  2. Find the work done by the pulling force in moving the block a distance of 5 m .
CAIE M1 2015 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-2_636_519_664_813} Three horizontal forces of magnitudes \(F \mathrm {~N} , 63 \mathrm {~N}\) and 25 N act at \(O\), the origin of the \(x\)-axis and \(y\)-axis. The forces are in equilibrium. The force of magnitude \(F \mathrm {~N}\) makes an angle \(\theta\) anticlockwise with the positive \(x\)-axis. The force of magnitude 63 N acts along the negative \(y\)-axis. The force of magnitude 25 N acts at \(\tan ^ { - 1 } 0.75\) clockwise from the negative \(x\)-axis (see diagram). Find the value of \(F\) and the value of \(\tan \theta\).
CAIE M1 2015 June Q3
3 A block of weight 6.1 N slides down a slope inclined at \(\tan ^ { - 1 } \left( \frac { 11 } { 60 } \right)\) to the horizontal. The coefficient of friction between the block and the slope is \(\frac { 1 } { 4 }\). The block passes through a point \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find how far the block moves from \(A\) before it comes to rest.
CAIE M1 2015 June Q4
4 A lorry of mass 14000 kg moves along a road starting from rest at a point \(O\). It reaches a point \(A\), and then continues to a point \(B\) which it reaches with a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The part \(O A\) of the road is straight and horizontal and has length 400 m . The part \(A B\) of the road is straight and is inclined downwards at an angle of \(\theta ^ { \circ }\) to the horizontal and has length 300 m .
  1. For the motion from \(O\) to \(B\), find the gain in kinetic energy of the lorry and express its loss in potential energy in terms of \(\theta\). The resistance to the motion of the lorry is 4800 N and the work done by the driving force of the lorry from \(O\) to \(B\) is 5000 kJ .
  2. Find the value of \(\theta\).
CAIE M1 2015 June Q5
5 A cyclist and her bicycle have a total mass of 84 kg . She works at a constant rate of \(P \mathrm {~W}\) while moving on a straight road which is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = 0.1\). When moving uphill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When moving downhill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the cyclist's motion, whether the cyclist is moving uphill or downhill, is \(R \mathrm {~N}\). Find the values of \(P\) and \(R\).
CAIE M1 2015 June Q6
6 Two particles \(A\) and \(B\) start to move at the same instant from a point \(O\). The particles move in the same direction along the same straight line. The acceleration of \(A\) at time \(t \mathrm {~s}\) after starting to move is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.05 - 0.0002 t\).
  1. Find A's velocity when \(t = 200\) and when \(t = 500\).
    \(B\) moves with constant acceleration for the first 200 s and has the same velocity as \(A\) when \(t = 200 . B\) moves with constant retardation from \(t = 200\) to \(t = 500\) and has the same velocity as \(A\) when \(t = 500\).
  2. Find the distance between \(A\) and \(B\) when \(t = 500\).
CAIE M1 2015 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-3_376_1052_1171_548} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a rough horizontal table with the string passing over a smooth pulley fixed at the edge of the table. The coefficient of friction between \(A\) and the table is 0.2 . Particle \(B\) hangs vertically below the pulley at a height of 0.5 m above the floor (see diagram). The system is released from rest and 0.25 s later the string breaks. A does not reach the pulley in the subsequent motion. Find
  1. the speed of \(B\) immediately before it hits the floor,
  2. the total distance travelled by \(A\).
CAIE M1 2015 June Q1
1 One end of a light inextensible string is attached to a block. The string makes an angle of \(60 ^ { \circ }\) above the horizontal and is used to pull the block in a straight line on a horizontal floor with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the string is 8 N . The block starts to move with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the first 5 s of the block's motion, find
  1. the distance travelled,
  2. the work done by the tension in the string.
CAIE M1 2015 June Q2
2 The total mass of a cyclist and his cycle is 80 kg . The resistance to motion is zero.
  1. The cyclist moves along a horizontal straight road working at a constant rate of \(P \mathrm {~W}\). Find the value of \(P\) given that the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when his acceleration is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. The cyclist moves up a straight hill inclined at an angle \(\alpha\), where \(\sin \alpha = 0.035\). Find the acceleration of the cyclist at an instant when he is working at a rate of 450 W and has speed \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2015 June Q3
3 A plane is inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 8 } \right)\) to the horizontal. \(A\) and \(B\) are two points on the same line of greatest slope with \(A\) higher than \(B\). The distance \(A B\) is 12 m . A small object \(P\) of mass 8 kg is released from rest at \(A\) and slides down the plane, passing through \(B\) with speed \(4.5 \mathrm {~ms} ^ { - 1 }\). For the motion of \(P\) from \(A\) to \(B\), find
  1. the increase in kinetic energy of \(P\) and the decrease in potential energy of \(P\),
  2. the magnitude of the constant resisting force that opposes the motion of \(P\).
CAIE M1 2015 June Q4
4 A particle \(P\) moves in a straight line. At time \(t\) seconds after starting from rest at the point \(O\) on the line, the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.075 t ^ { 2 } - 1.5 t + 5\).
  1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
  2. Hence find the time taken for \(P\) to return to the point \(O\).
CAIE M1 2015 June Q5
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
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
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
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
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
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
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
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
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
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
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
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
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.