Questions M1 (1912 questions)

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CAIE M1 2012 June Q4
4 A particle \(P\) starts at the point \(O\) and travels in a straight line. At time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.75 t ^ { 2 } - 0.0625 t ^ { 3 }\). Find
  1. the positive value of \(t\) for which the acceleration is zero,
  2. the distance travelled by \(P\) before it changes its direction of motion.
CAIE M1 2012 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_485_874_255_638} The diagram shows the vertical cross-section \(O A B\) of a slide. The straight line \(A B\) is tangential to the curve \(O A\) at \(A\). The line \(A B\) is inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The point \(O\) is 10 m higher than \(B\), and \(A B\) has length 10 m (see diagram). The part of the slide containing the curve \(O A\) is smooth and the part containing \(A B\) is rough. A particle \(P\) of mass 2 kg is released from rest at \(O\) and moves down the slide.
  1. Find the speed of \(P\) when it passes through \(A\). The coefficient of friction between \(P\) and the part of the slide containing \(A B\) is \(\frac { 1 } { 12 }\). Find
  2. the acceleration of \(P\) when it is moving from \(A\) to \(B\),
  3. the speed of \(P\) when it reaches \(B\).
CAIE M1 2012 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_465_849_1475_648} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a vertical cross-section of a triangular prism. The base of the prism is fixed on horizontal ground and each of the sloping sides is smooth. Each sloping side makes an angle \(\theta\) with the ground, where \(\sin \theta = 0.8\). Initially the particles are held at rest on the sloping sides, with the string taut (see diagram). The particles are released and move along lines of greatest slope.
  1. Find the tension in the string and the acceleration of the particles while both are moving. The speed of \(P\) when it reaches the ground is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). On reaching the ground \(P\) comes to rest and remains at rest. \(Q\) continues to move up the slope but does not reach the pulley.
  2. Find the time taken from the instant that the particles are released until \(Q\) reaches its greatest height above the ground.
CAIE M1 2012 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-4_529_481_255_831} A small ring of mass 0.2 kg is threaded on a fixed vertical rod. The end \(A\) of a light inextensible string is attached to the ring. The other end \(C\) of the string is attached to a fixed point of the rod above \(A\). A horizontal force of magnitude 8 N is applied to the point \(B\) of the string, where \(A B = 1.5 \mathrm {~m}\) and \(B C = 2 \mathrm {~m}\). The system is in equilibrium with the string taut and \(A B\) at right angles to \(B C\) (see diagram).
  1. Find the tension in the part \(A B\) of the string and the tension in the part \(B C\) of the string. The equilibrium is limiting with the ring on the point of sliding up the rod.
  2. Find the coefficient of friction between the ring and the rod.
CAIE M1 2012 June Q1
1 A block is pulled in a straight line along horizontal ground by a force of constant magnitude acting at an angle of \(60 ^ { \circ }\) above the horizontal. The work done by the force in moving the block a distance of 5 m is 75 J . Find the magnitude of the force.
CAIE M1 2012 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_465_478_479_836} Three coplanar forces of magnitudes \(F \mathrm {~N} , 12 \mathrm {~N}\) and 15 N are in equilibrium acting at a point \(P\) in the directions shown in the diagram. Find \(\alpha\) and \(F\).
CAIE M1 2012 June Q3
3 A particle \(P\) moves in a straight line, starting from the point \(O\) with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(2 t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that \(t ^ { \frac { 5 } { 3 } } = \frac { 5 } { 6 }\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the distance of \(P\) from \(O\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2012 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_168_711_1612_717} A ring of mass 4 kg is attached to one end of a light string. The ring is threaded on a fixed horizontal rod and the string is pulled at an angle of \(25 ^ { \circ }\) below the horizontal (see diagram). With a tension in the string of \(T \mathrm {~N}\) the ring is in equilibrium.
  1. Find, in terms of \(T\), the horizontal and vertical components of the force exerted on the ring by the rod. The coefficient of friction between the ring and the rod is 0.4 .
  2. Given that the equilibrium is limiting, find the value of \(T\).
CAIE M1 2012 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_529_195_255_977} A block \(A\) of mass 3 kg is attached to one end of a light inextensible string \(S _ { 1 }\). Another block \(B\) of mass 2 kg is attached to the other end of \(S _ { 1 }\), and is also attached to one end of another light inextensible string \(S _ { 2 }\). The other end of \(S _ { 2 }\) is attached to a fixed point \(O\) and the blocks hang in equilibrium below \(O\) (see diagram).
  1. Find the tension in \(S _ { 1 }\) and the tension in \(S _ { 2 }\). The string \(S _ { 2 }\) breaks and the particles fall. The air resistance on \(A\) is 1.6 N and the air resistance on \(B\) is 4 N .
  2. Find the acceleration of the particles and the tension in \(S _ { 1 }\).
CAIE M1 2012 June Q6
6 A car of mass 1250 kg travels from the bottom to the top of a straight hill which has length 400 m and is inclined to the horizontal at an angle of \(\alpha\), where \(\sin \alpha = 0.125\). The resistance to the car's motion is 800 N . Find the work done by the car's engine in each of the following cases.
  1. The car's speed is constant.
  2. The car's initial speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car's driving force is 3 times greater at the top of the hill than it is at the bottom, and the car's power output is 5 times greater at the top of the hill than it is at the bottom.
CAIE M1 2012 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_168_803_1909_671} The frictional force acting on a small block of mass 0.15 kg , while it is moving on a horizontal surface, has magnitude 0.12 N . The block is set in motion from a point \(X\) on the surface, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits a vertical wall at a point \(Y\) on the surface 2 s later. The block rebounds from the wall and moves directly towards \(X\) before coming to rest at the point \(Z\) (see diagram). At the instant that the block hits the wall it loses 0.072 J of its kinetic energy. The velocity of the block, in the direction from \(X\) to \(Y\), is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after it leaves \(X\).
  1. Find the values of \(v\) when the block arrives at \(Y\) and when it leaves \(Y\), and find also the value of \(t\) when the block comes to rest at \(Z\). Sketch the velocity-time graph.
  2. The displacement of the block from \(X\), in the direction from \(X\) to \(Y\), is \(s \mathrm {~m}\) at time \(t \mathrm {~s}\). Sketch the displacement-time graph. Show on your graph the values of \(s\) and \(t\) when the block is at \(Y\) and when it comes to rest at \(Z\).
CAIE M1 2012 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-2_262_711_248_717} A ring is threaded on a fixed horizontal bar. The ring is attached to one end of a light inextensible string which is used to pull the ring along the bar at a constant speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The string makes a constant angle of \(24 ^ { \circ }\) with the bar and the tension in the string is 6 N (see diagram). Find the work done by the tension in a period of 8 s .
CAIE M1 2012 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-2_471_621_870_762} A smooth ring \(R\) of mass 0.16 kg is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\). A horizontal force of magnitude 11.2 N acts on \(R\), in the same vertical plane as \(A\) and \(B\). The ring is in equilibrium. The string is taut with angle \(A R B = 90 ^ { \circ }\), and the part \(A R\) of the string makes an angle of \(\theta ^ { \circ }\) with the horizontal (see diagram). The tension in the string is \(T \mathrm {~N}\).
  1. Find two simultaneous equations involving \(T \sin \theta\) and \(T \cos \theta\).
  2. Hence find \(T\) and \(\theta\).
CAIE M1 2012 June Q3
3 A particle \(P\) travels from a point \(O\) along a straight line and comes to instantaneous rest at a point \(A\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.027 \left( 10 t ^ { 2 } - t ^ { 3 } \right)\). Find
  1. the distance \(O A\),
  2. the maximum velocity of \(P\) while moving from \(O\) to \(A\).
CAIE M1 2012 June Q4
4 A car of mass 1230 kg increases its speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 24.5 s . The table below shows corresponding values of time \(t \mathrm {~s}\) and speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
\(t\)00.516.324.5
\(v\)461921
  1. Using the values in the table, find the average acceleration of the car for \(0 < t < 0.5\) and for \(16.3 < t < 24.5\). While the car is increasing its speed the power output of its engine is constant and equal to \(P \mathrm {~W}\), and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\).
  2. Assuming that the values obtained in part (i) are approximately equal to the accelerations at \(v = 5\) and at \(v = 20\), find approximations for \(P\) and \(R\).
CAIE M1 2012 June Q5
4 marks
5 A lorry of mass 16000 kg moves on a straight hill inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The length of the hill is 500 m .
  1. While the lorry moves from the bottom to the top of the hill at constant speed, the resisting force acting on the lorry is 800 N and the work done by the driving force is 2800 kJ . Find the value of \(\alpha\).
  2. On the return journey the speed of the lorry is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill. While the lorry travels down the hill, the work done by the driving force is 2400 kJ and the work done against the resistance to motion is 800 kJ . Find the speed of the lorry at the bottom of the hill.
    [0pt] [4]
CAIE M1 2012 June Q6
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918b65cc-617d-4942-8d96-b02eef21e417-3_156_558_1592_450} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918b65cc-617d-4942-8d96-b02eef21e417-3_138_559_1612_1137} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A block of weight 6.1 N is at rest on a plane inclined at angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 11 } { 60 }\). The coefficient of friction between the block and the plane is \(\mu\). A force of magnitude 5.9 N acting parallel to a line of greatest slope is applied to the block.
  1. When the force acts up the plane (see Fig. 1) the block remains at rest. Show that \(\mu \geqslant \frac { 4 } { 5 }\).
  2. When the force acts down the plane (see Fig. 2) the block slides downwards. Show that \(\mu < \frac { 7 } { 6 }\).
  3. Given that the acceleration of the block is \(1.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when the force acts down the plane, find the value of \(\mu\).
CAIE M1 2012 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-4_506_471_255_836} Two particles \(A\) and \(B\) have masses 0.12 kg and 0.38 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest with the string taut and both straight parts of the string vertical. \(A\) and \(B\) are each at a height of 0.65 m above horizontal ground (see diagram). \(A\) is released and \(B\) moves downwards. Find
  1. the acceleration of \(B\) while it is moving downwards,
  2. the speed with which \(B\) reaches the ground and the time taken for it to reach the ground.
    \(B\) remains on the ground while \(A\) continues to move with the string slack, without reaching the pulley. The string remains slack until \(A\) is at a height of 1.3 m above the ground for a second time. At this instant \(A\) has been in motion for a total time of \(T \mathrm {~s}\).
  3. Find the value of \(T\) and sketch the velocity-time graph for \(A\) for the first \(T \mathrm {~s}\) of its motion.
  4. Find the total distance travelled by \(A\) in the first \(T\) s of its motion.
CAIE M1 2013 June Q1
1 A block is at rest on a rough horizontal plane. The coefficient of friction between the block and the plane is 1.25 .
  1. State, giving a reason for your answer, whether the minimum vertical force required to move the block is greater or less than the minimum horizontal force required to move the block. A horizontal force of continuously increasing magnitude \(P \mathrm {~N}\) and fixed direction is applied to the block.
  2. Given that the weight of the block is 60 N , find the value of \(P\) when the acceleration of the block is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2013 June Q2
2 A car of mass 1250 kg travels from the bottom to the top of a straight hill of length 600 m , which is inclined at an angle of \(2.5 ^ { \circ }\) to the horizontal. The resistance to motion of the car is constant and equal to 400 N . The work done by the driving force is 450 kJ . The speed of the car at the bottom of the hill is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of the car at the top of the hill.
CAIE M1 2013 June Q3
3 The top of a cliff is 40 metres above the level of the sea. A man in a boat, close to the bottom of the cliff, is in difficulty and fires a distress signal vertically upwards from sea level. Find
  1. the speed of projection of the signal given that it reaches a height of 5 m above the top of the cliff,
  2. the length of time for which the signal is above the level of the top of the cliff. The man fires another distress signal vertically upwards from sea level. This signal is above the level of the top of the cliff for \(\sqrt { } ( 17 ) \mathrm { s }\).
  3. Find the speed of projection of the second signal.
CAIE M1 2013 June Q4
4 A train of mass 400000 kg is moving on a straight horizontal track. The power of the engine is constant and equal to 1500 kW and the resistance to the train's motion is 30000 N . Find
  1. the acceleration of the train when its speed is \(37.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  2. the steady speed at which the train can move.
CAIE M1 2013 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_275_663_258_742} A light inextensible string has a particle \(A\) of mass 0.26 kg attached to one end and a particle \(B\) of mass 0.54 kg attached to the other end. The particle \(A\) is held at rest on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). The string is taut and parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley at the top of the plane. Particle \(B\) hangs at rest vertically below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is 0.2 . Particle \(A\) is released and the particles start to move.
  1. Find the magnitude of the acceleration of the particles and the tension in the string. Particle \(A\) reaches the pulley 0.4 s after starting to move.
  2. Find the distance moved by each of the particles.
CAIE M1 2013 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_639_939_1260_603} A particle \(P\) of mass 0.5 kg lies on a smooth horizontal plane. Horizontal forces of magnitudes \(F \mathrm {~N}\), 2.5 N and 2.6 N act on \(P\). The directions of the forces are as shown in the diagram, where \(\tan \alpha = \frac { 12 } { 5 }\) and \(\tan \beta = \frac { 7 } { 24 }\).
  1. Given that \(P\) is in equilibrium, find the values of \(F\) and \(\tan \theta\).
  2. The force of magnitude \(F \mathrm {~N}\) is removed. Find the magnitude and direction of the acceleration with which \(P\) starts to move.
CAIE M1 2013 June Q7
7 A car driver makes a journey in a straight line from \(A\) to \(B\), starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at \(B\). The distance travelled by the car \(t\) seconds after leaving \(A\) is \(0.0000117 \left( 400 t ^ { 3 } - 3 t ^ { 4 } \right)\) metres.
  1. Find the distance \(A B\).
  2. Find the maximum speed of the car.
  3. Find the acceleration of the car
    (a) as it starts from \(A\),
    (b) as it arrives at \(B\).
  4. Sketch the velocity-time graph for the journey.