Questions M1 (1912 questions)

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CAIE M1 2008 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{a4cb105b-55d2-4793-95d2-3d791990a1f6-3_643_481_274_831} Particles \(A\) and \(B\), of masses 0.5 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held at rest on the horizontal floor and particle \(A\) hangs in equilibrium (see diagram). \(B\) is released and each particle starts to move vertically. \(A\) hits the floor 2 s after \(B\) is released. The speed of each particle when \(A\) hits the floor is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. For the motion while \(A\) is moving downwards, find
    (a) the acceleration of \(A\),
    (b) the tension in the string.
  2. Find the value of \(m\).
CAIE M1 2008 November Q6
6 A train travels from \(A\) to \(B\), a distance of 20000 m , taking 1000 s . The journey has three stages. In the first stage the train starts from rest at \(A\) and accelerates uniformly until its speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the second stage the train travels at constant speed \(V _ { \mathrm { m } } { } ^ { - 1 }\) for 600 s . During the third stage of the journey the train decelerates uniformly, coming to rest at \(B\).
  1. Sketch the velocity-time graph for the train's journey.
  2. Find the value of \(V\).
  3. Given that the acceleration of the train during the first stage of the journey is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the distance travelled by the train during the third stage of the journey.
    \(7 \quad\) A particle \(P\) is held at rest at a fixed point \(O\) and then released. \(P\) falls freely under gravity until it reaches the point \(A\) which is 1.25 m below \(O\).
  4. Find the speed of \(P\) at \(A\) and the time taken for \(P\) to reach \(A\). The particle continues to fall, but now its downward acceleration \(t\) seconds after passing through \(A\) is \(( 10 - 0.3 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  5. Find the total distance \(P\) has fallen, 3 s after being released from \(O\).
CAIE M1 2009 November Q1
1 A car of mass 1000 kg moves along a horizontal straight road, passing through points \(A\) and \(B\). The power of its engine is constant and equal to 15000 W . The driving force exerted by the engine is 750 N at \(A\) and 500 N at \(B\). Find the speed of the car at \(A\) and at \(B\), and hence find the increase in the car's kinetic energy as it moves from \(A\) to \(B\).
CAIE M1 2009 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-2_609_967_536_589} A smooth narrow tube \(A E\) has two straight parts, \(A B\) and \(D E\), and a curved part \(B C D\). The part \(A B\) is vertical with \(A\) above \(B\), and \(D E\) is horizontal. \(C\) is the lowest point of the tube and is 0.65 m below the level of \(D E\). A particle is released from rest at \(A\) and travels through the tube, leaving it at \(E\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) (see diagram). Find
  1. the height of \(A\) above the level of \(D E\),
  2. the maximum speed of the particle.
CAIE M1 2009 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-2_462_721_1672_712} Two forces have magnitudes \(P \mathrm {~N}\) and \(Q \mathrm {~N}\). The resultant of the two forces has magnitude 12 N and acts in a direction \(40 ^ { \circ }\) clockwise from the force of magnitude \(P \mathrm {~N}\) and \(80 ^ { \circ }\) anticlockwise from the force of magnitude \(Q \mathrm {~N}\) (see diagram). Find the value of \(Q\).
CAIE M1 2009 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-3_335_751_264_696} A particle \(P\) of weight 5 N is attached to one end of each of two light inextensible strings of lengths 30 cm and 40 cm . The other end of the shorter string is attached to a fixed point \(A\) of a rough rod which is fixed horizontally. A small ring \(S\) of weight \(W \mathrm {~N}\) is attached to the other end of the longer string and is threaded on to the rod. The system is in equilibrium with the strings taut and \(A S = 50 \mathrm {~cm}\) (see diagram).
  1. By resolving the forces acting on \(P\) in the direction of \(P S\), or otherwise, find the tension in the longer string.
  2. Find the magnitude of the frictional force acting on \(S\).
  3. Given that the coefficient of friction between \(S\) and the rod is 0.75 , and that \(S\) is in limiting equilibrium, find the value of \(W\).
CAIE M1 2009 November Q5
5 A particle \(P\) of mass 0.6 kg moves upwards along a line of greatest slope of a plane inclined at \(18 ^ { \circ }\) to the horizontal. The deceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the frictional and normal components of the force exerted on \(P\) by the plane. Hence find the coefficient of friction between \(P\) and the plane, correct to 2 significant figures. After \(P\) comes to instantaneous rest it starts to move down the plane with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(a\).
CAIE M1 2009 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-4_712_529_264_810} Particles \(P\) and \(Q\), of masses 0.55 kg and 0.45 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The particles are held at rest with the string taut and its straight parts vertical. Both particles are at a height of 5 m above the ground (see diagram). The system is released.
  1. Find the acceleration with which \(P\) starts to move. The string breaks after 2 s and in the subsequent motion \(P\) and \(Q\) move vertically under gravity.
  2. At the instant that the string breaks, find
    (a) the height above the ground of \(P\) and of \(Q\),
    (b) the speed of the particles.
  3. Show that \(Q\) reaches the ground 0.8 s later than \(P\).
    \(7 \quad\) A particle \(P\) starts from rest at the point \(A\) at time \(t = 0\), where \(t\) is in seconds, and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 10 s . For \(10 \leqslant t \leqslant 20 , P\) continues to move along the line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 800 } { t ^ { 2 } } - 2\). Find
  4. the speed of \(P\) when \(t = 10\), and the value of \(a\),
  5. the value of \(t\) for which the acceleration of \(P\) is \(- a \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
  6. the displacement of \(P\) from \(A\) when \(t = 20\). \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 2009 November Q1
1 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{efa7175f-832b-4cd3-82ab-52e402115081-2_458_472_267_493} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{efa7175f-832b-4cd3-82ab-52e402115081-2_351_435_365_1217} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small block of weight 12 N is at rest on a smooth plane inclined at \(40 ^ { \circ }\) to the horizontal. The block is held in equilibrium by a force of magnitude \(P \mathrm {~N}\). Find the value of \(P\) when
  1. the force is parallel to the plane as in Fig. 1,
  2. the force is horizontal as in Fig. 2.
CAIE M1 2009 November Q2
2 A lorry of mass 15000 kg moves with constant speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top to the bottom of a straight hill of length 900 m . The top of the hill is 18 m above the level of the bottom of the hill. The total work done by the resistive forces acting on the lorry, including the braking force, is \(4.8 \times 10 ^ { 6 } \mathrm {~J}\). Find
  1. the loss in gravitational potential energy of the lorry,
  2. the work done by the driving force. On reaching the bottom of the hill the lorry continues along a straight horizontal road against a constant resistance of 1600 N . There is no braking force acting. The speed of the lorry increases from \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill to \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(X\), where \(X\) is 2500 m from the bottom of the hill.
  3. By considering energy, find the work done by the driving force of the lorry while it travels from the bottom of the hill to \(X\).
CAIE M1 2009 November Q3
3 A car of mass 1250 kg travels along a horizontal straight road with increasing speed. The power provided by the car's engine is constant and equal to 24 kW . The resistance to the car's motion is constant and equal to 600 N .
  1. Show that the speed of the car cannot exceed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the acceleration of the car at an instant when its speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2009 November Q4
4 A particle moves up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\) and \(\sin \alpha = 0.28\).
  1. Given that the normal component of the contact force acting on the particle has magnitude 1.2 N , find the mass of the particle.
  2. Given also that the frictional component of the contact force acting on the particle has magnitude 0.4 N , find the deceleration of the particle. The particle comes to rest on reaching the point \(X\).
  3. Determine whether the particle remains at \(X\) or whether it starts to move down the plane.
CAIE M1 2009 November Q5
5 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{efa7175f-832b-4cd3-82ab-52e402115081-3_317_517_922_468} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{efa7175f-832b-4cd3-82ab-52e402115081-3_317_522_922_1155} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small ring of weight 12 N is threaded on a fixed rough horizontal rod. A light string is attached to the ring and the string is pulled with a force of 15 N at an angle of \(30 ^ { \circ }\) to the horizontal.
  1. When the angle of \(30 ^ { \circ }\) is below the horizontal (see Fig. 1), the ring is in limiting equilibrium. Show that the coefficient of friction between the ring and the rod is 0.666 , correct to 3 significant figures.
  2. When the angle of \(30 ^ { \circ }\) is above the horizontal (see Fig. 2), the ring is moving with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(a\).
CAIE M1 2009 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{efa7175f-832b-4cd3-82ab-52e402115081-4_686_511_269_817} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is held on the horizontal floor and particle \(B\) hangs in equilibrium. Particle \(A\) is released and both particles start to move vertically.
  1. Find the acceleration of the particles. The speed of the particles immediately before \(B\) hits the floor is \(1.6 \mathrm {~ms} ^ { - 1 }\). Given that \(B\) does not rebound upwards, find
  2. the maximum height above the floor reached by \(A\),
  3. the time taken by \(A\), from leaving the floor, to reach this maximum height.
CAIE M1 2009 November Q7
7 A motorcyclist starts from rest at \(A\) and travels in a straight line. For the first part of the motion, the motorcyclist's displacement \(x\) metres from \(A\) after \(t\) seconds is given by \(x = 0.6 t ^ { 2 } - 0.004 t ^ { 3 }\).
  1. Show that the motorcyclist's acceleration is zero when \(t = 50\) and find the speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at this time. For \(t \geqslant 50\), the motorcyclist travels at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(t\) for which the motorcyclist's average speed is \(27.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \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 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_608_831_258_657} Two particles \(P\) and \(Q\) move vertically under gravity. The graphs show the upward velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particles at time \(t \mathrm {~s}\), for \(0 \leqslant t \leqslant 4 . P\) starts with velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) starts from rest.
  1. Find the value of \(V\). Given that \(Q\) reaches the horizontal ground when \(t = 4\), find
  2. the speed with which \(Q\) reaches the ground,
  3. the height of \(Q\) above the ground when \(t = 0\).
CAIE M1 2010 November Q2
2 A car of mass 600 kg travels along a horizontal straight road, with its engine working at a rate of 40 kW . The resistance to motion of the car is constant and equal to 800 N . The car passes through the point \(A\) on the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's acceleration at the point \(B\) on the road is half its acceleration at \(A\). Find the speed of the car at \(B\).
CAIE M1 2010 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_606_843_1731_651} The diagram shows three particles \(A , B\) and \(C\) hanging freely in equilibrium, each being attached to the end of a string. The other ends of the three strings are tied together and are at the point \(X\). The strings carrying \(A\) and \(C\) pass over smooth fixed horizontal pegs \(P _ { 1 }\) and \(P _ { 2 }\) respectively. The weights of \(A , B\) and \(C\) are \(5.5 \mathrm {~N} , 7.3 \mathrm {~N}\) and \(W \mathrm {~N}\) respectively, and the angle \(P _ { 1 } X P _ { 2 }\) is a right angle. Find the angle \(A P _ { 1 } X\) and the value of \(W\).
CAIE M1 2010 November Q4
4 A particle \(P\) starts from a fixed point \(O\) at time \(t = 0\), where \(t\) is in seconds, and moves with constant acceleration in a straight line. The initial velocity of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its velocity when \(t = 10\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the displacement of \(P\) from \(O\) when \(t = 10\). Another particle \(Q\) also starts from \(O\) when \(t = 0\) and moves along the same straight line as \(P\). The acceleration of \(Q\) at time \(t\) is \(0.03 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that \(Q\) has the same velocity as \(P\) when \(t = 10\), show that it also has the same displacement from \(O\) as \(P\) when \(t = 10\).
CAIE M1 2010 November Q5
5 A particle of mass 0.8 kg slides down a rough inclined plane along a line of greatest slope \(A B\). The distance \(A B\) is 8 m . The particle starts at \(A\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves with constant acceleration \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the speed of the particle at the instant it reaches \(B\).
  2. Given that the work done against the frictional force as the particle moves from \(A\) to \(B\) is 7 J , find the angle of inclination of the plane. When the particle is at the point \(X\) its speed is the same as the average speed for the motion from \(A\) to \(B\).
  3. Find the work done by the frictional force for the particle's motion from \(A\) to \(X\).
CAIE M1 2010 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-3_476_1305_1519_420} A smooth slide \(A B\) is fixed so that its highest point \(A\) is 3 m above horizontal ground. \(B\) is \(h \mathrm {~m}\) above the ground. A particle \(P\) of mass 0.2 kg is released from rest at a point on the slide. The particle moves down the slide and, after passing \(B\), continues moving until it hits the ground (see diagram). The speed of \(P\) at \(B\) is \(v _ { B }\) and the speed at which \(P\) hits the ground is \(v _ { G }\).
  1. In the case that \(P\) is released at \(A\), it is given that the kinetic energy of \(P\) at \(B\) is 1.6 J . Find
    (a) the value of \(h\),
    (b) the kinetic energy of the particle immediately before it reaches the ground,
    (c) the ratio \(v _ { G } : v _ { B }\).
  2. In the case that \(P\) is released at the point \(X\) of the slide, which is \(H \mathrm {~m}\) above the ground (see diagram), it is given that \(v _ { G } : v _ { B } = 2.55\). Find the value of \(H\) correct to 2 significant figures.
    \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-4_384_679_258_733} Particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(P\) hangs freely and \(Q\) is in contact with the table. A force of magnitude 3.2 N acts on \(Q\), upwards and away from the pulley, at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).
  3. The system is in limiting equilibrium with \(P\) about to move upwards. Find the coefficient of friction between \(Q\) and the table. The force of magnitude 3.2 N is now removed and \(P\) starts to move downwards.
  4. Find the acceleration of the particles and the tension in the string. \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 November Q1
1 A block of mass 400 kg rests in limiting equilibrium on horizontal ground. A force of magnitude 2000 N acts on the block at an angle of \(15 ^ { \circ }\) to the upwards vertical. Find the coefficient of friction between the block and the ground, correct to 2 significant figures.
CAIE M1 2010 November Q2
2 A cyclist, working at a constant rate of 400 W , travels along a straight road which is inclined at \(2 ^ { \circ }\) to the horizontal. The total mass of the cyclist and his cycle is 80 kg . Ignoring any resistance to motion, find, correct to 1 decimal place, the acceleration of the cyclist when he is travelling
  1. uphill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  2. downhill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2010 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-2_597_616_888_762} A particle \(P\) is in equilibrium on a smooth horizontal table under the action of four horizontal forces of magnitudes \(6 \mathrm {~N} , 5 \mathrm {~N} , F \mathrm {~N}\) and \(F \mathrm {~N}\) acting in the directions shown. Find the values of \(\alpha\) and \(F\).
CAIE M1 2010 November Q4
4 A block of mass 20 kg is pulled from the bottom to the top of a slope. The slope has length 10 m and is inclined at \(4.5 ^ { \circ }\) to the horizontal. The speed of the block is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the slope and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the slope.
  1. Find the loss of kinetic energy and the gain in potential energy of the block.
  2. Given that the work done against the resistance to motion is 50 J , find the work done by the pulling force acting on the block.
  3. Given also that the pulling force is constant and acts at an angle of \(15 ^ { \circ }\) upwards from the slope, find its magnitude.