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CAIE M1 2006 June Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_305_599_1717_774} Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string. \(P\) is at rest on a rough horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. \(Q\) hangs vertically below the pulley (see diagram). The force exerted on the string by the pulley has magnitude \(4 \sqrt { } 2 \mathrm {~N}\). The coefficient of friction between \(P\) and the table is 0.8 .
  1. Show that the tension in the string is 4 N and state the mass of \(Q\).
  2. Given that \(P\) is on the point of slipping, find its mass. A particle of mass 0.1 kg is now attached to \(Q\) and the system starts to move.
  3. Find the tension in the string while the particles are in motion.
CAIE M1 2006 June Q6
9 marks Standard +0.3
6 A block of mass 50 kg is pulled up a straight hill and passes through points \(A\) and \(B\) with speeds \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 200 m and \(B\) is 15 m higher than \(A\). For the motion of the block from \(A\) to \(B\), find
  1. the loss in kinetic energy of the block,
  2. the gain in potential energy of the block. The resistance to motion of the block has magnitude 7.5 N.
  3. Find the work done by the pulling force acting on the block. The pulling force acting on the block has constant magnitude 45 N and acts at an angle \(\alpha ^ { \circ }\) upwards from the hill.
  4. Find the value of \(\alpha\).
CAIE M1 2006 June Q7
10 marks Standard +0.3
7 Two particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. The particles start at the same instant and from the same point, each with speed \(1.3 \mathrm {~ms} ^ { - 1 }\). Initially \(P\) moves down the plane and \(Q\) moves up the plane. The distance between the particles \(t\) seconds after they start to move is \(d \mathrm {~m}\).
  1. Show that \(d = 2.6 t\). When \(t = 2.5\) the difference in the vertical height of the particles is 1.6 m . Find
  2. the acceleration of the particles down the plane,
  3. the distance travelled by \(P\) when \(Q\) is at its highest point.
CAIE M1 2007 June Q1
4 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-2_203_1200_264_475} A particle slides up a line of greatest slope of a smooth plane inclined at an angle \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the points \(A\) and \(B\) with speeds \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 4 m (see diagram). Find
  1. the deceleration of the particle,
  2. the value of \(\alpha\).
CAIE M1 2007 June Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-2_549_589_934_778} Two forces, each of magnitude 8 N , act at a point in the directions \(O A\) and \(O B\). The angle between the forces is \(\theta ^ { \circ }\) (see diagram). The resultant of the two forces has component 9 N in the direction \(O A\). Find
  1. the value of \(\theta\),
  2. the magnitude of the resultant of the two forces.
CAIE M1 2007 June Q3
6 marks Moderate -0.5
3 A car travels along a horizontal straight road with increasing speed until it reaches its maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and equal to \(R \mathrm {~N}\), and the power provided by the car's engine is 18 kW .
  1. Find the value of \(R\).
  2. Given that the car has mass 1200 kg , find its acceleration at the instant when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2007 June Q4
7 marks Moderate -0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-3_702_709_269_719} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed peg. The particles are held at rest with the string taut. Both particles are at a height of 0.9 m above the ground (see diagram). The system is released and each of the particles moves vertically. Find
  1. the acceleration of \(P\) and the tension in the string before \(P\) reaches the ground,
  2. the time taken for \(P\) to reach the ground.
CAIE M1 2007 June Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-3_223_1456_1493_347} A lorry of mass 12500 kg travels along a road that has a straight horizontal section \(A B\) and a straight inclined section \(B C\). The length of \(B C\) is 500 m . The speeds of the lorry at \(A , B\) and \(C\) are \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).
  1. The work done against the resistance to motion of the lorry, as it travels from \(A\) to \(B\), is 5000 kJ . Find the work done by the driving force as the lorry travels from \(A\) to \(B\).
  2. As the lorry travels from \(B\) to \(C\), the resistance to motion is 4800 N and the work done by the driving force is 3300 kJ . Find the height of \(C\) above the level of \(A B\).
CAIE M1 2007 June Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_593_746_269_701} A particle \(P\) starts from rest at the point \(A\) and travels in a straight line, coming to rest again after 10 s . The velocity-time graph for \(P\) consists of two straight line segments (see diagram). A particle \(Q\) starts from rest at \(A\) at the same instant as \(P\) and travels along the same straight line as \(P\). The velocity of \(Q\) is given by \(v = 3 t - 0.3 t ^ { 2 }\) for \(0 \leqslant t \leqslant 10\). The displacements from \(A\) of \(P\) and \(Q\) are the same when \(t = 10\).
  1. Show that the greatest velocity of \(P\) during its motion is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(t\), in the interval \(0 < t < 5\), for which the acceleration of \(Q\) is the same as the acceleration of \(P\).
CAIE M1 2007 June Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_414_865_1512_641} Two light strings are attached to a block of mass 20 kg . The block is in equilibrium on a horizontal surface \(A B\) with the strings taut. The strings make angles of \(60 ^ { \circ }\) and \(30 ^ { \circ }\) with the horizontal, on either side of the block, and the tensions in the strings are \(T \mathrm {~N}\) and 75 N respectively (see diagram).
  1. Given that the surface is smooth, find the value of \(T\) and the magnitude of the contact force acting on the block.
  2. It is given instead that the surface is rough and that the block is on the point of slipping. The frictional force on the block has magnitude 25 N and acts towards \(A\). Find the coefficient of friction between the block and the surface. \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 2008 June Q1
4 marks Moderate -0.8
1 A particle slides down a smooth plane inclined at an angle of \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the point \(A\) with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 1.2 s later it passes through the point \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the acceleration of the particle,
  2. the value of \(\alpha\).
CAIE M1 2008 June Q2
4 marks Moderate -0.8
2 A block is being pulled along a horizontal floor by a rope inclined at \(20 ^ { \circ }\) to the horizontal. The tension in the rope is 851 N and the block moves at a constant speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the work done on the block in 12 s is approximately 24 kJ .
  2. Hence find the power being applied to the block, giving your answer to the nearest kW .
CAIE M1 2008 June Q3
5 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-2_520_565_1009_792} Three horizontal forces of magnitudes \(F \mathrm {~N} , 13 \mathrm {~N}\) and 10 N act at a fixed point \(O\) and are in equilibrium. The directions of the forces are as shown in the diagram. Find, in either order, the value of \(\theta\) and the value of \(F\).
CAIE M1 2008 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_478_1041_269_552} \(O A B C\) is a vertical cross-section of a smooth surface. The straight part \(O A\) has length 2.4 m and makes an angle of \(50 ^ { \circ }\) with the horizontal. \(A\) and \(C\) are at the same horizontal level and \(B\) is the lowest point of the cross-section (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(O\) and moves on the surface. \(P\) remains in contact with the surface until it leaves the surface at \(C\). Find
  1. the kinetic energy of \(P\) at \(A\),
  2. the speed of \(P\) at \(C\). The greatest speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the depth of \(B\) below the horizontal through \(A\) and \(C\).
CAIE M1 2008 June Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_314_867_1457_639} A block \(B\) of mass 0.6 kg and a particle \(A\) of mass 0.4 kg are attached to opposite ends of a light inextensible string. The block is held at rest on a rough horizontal table, and the coefficient of friction between the block and the table is 0.5 . The string passes over a small smooth pulley \(C\) at the edge of the table and \(A\) hangs in equilibrium vertically below \(C\). The part of the string between \(B\) and \(C\) is horizontal and the distance \(B C\) is 3 m (see diagram). \(B\) is released and the system starts to move.
  1. Find the acceleration of \(B\) and the tension in the string.
  2. Find the time taken for \(B\) to reach the pulley.
CAIE M1 2008 June Q6
9 marks Standard +0.8
6 A particle \(P\) of mass 0.6 kg is projected vertically upwards with speed \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) which is 6.2 m above the ground. Air resistance acts on \(P\) so that its deceleration is \(10.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving upwards, and its acceleration is \(9.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving downwards. Find
  1. the greatest height above the ground reached by \(P\),
  2. the speed with which \(P\) reaches the ground,
  3. the total work done on \(P\) by the air resistance.
CAIE M1 2008 June Q7
13 marks Moderate -0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-4_719_1059_264_543} An object \(P\) travels from \(A\) to \(B\) in a time of 80 s . The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\). The graph consists of straight line segments for the intervals \(0 \leqslant t \leqslant 10\) and \(30 \leqslant t \leqslant 80\), and a curved section whose equation is \(v = - 0.01 t ^ { 2 } + 0.5 t - 1\) for \(10 \leqslant t \leqslant 30\). Find
  1. the maximum velocity of \(P\),
  2. the distance \(A B\). \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 June Q1
3 marks Easy -1.3
1 \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_388_565_264_790} A block \(B\) of mass 5 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 4 kg is attached to other end of the string. The string passes over a smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on the ground (see diagram). State the tension in the string and find the force exerted on \(B\) by the ground.
CAIE M1 2009 June Q2
3 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_349_1139_1011_502} A crate \(C\) is pulled at constant speed up a straight inclined path by a constant force of magnitude \(F \mathrm {~N}\), acting upwards at an angle of \(15 ^ { \circ }\) to the path. \(C\) passes through points \(P\) and \(Q\) which are 100 m apart (see diagram). As \(C\) travels from \(P\) to \(Q\) the work done against the resistance to \(C\) 's motion is 900 J , and the gain in \(C\) 's potential energy is 2100 J . Write down the work done by the pulling force as \(C\) travels from \(P\) to \(Q\), and hence find the value of \(F\).
CAIE M1 2009 June Q3
5 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_492_606_1763_772} Forces of magnitudes \(7 \mathrm {~N} , 10 \mathrm {~N}\) and 15 N act on a particle in the directions shown in the diagram.
  1. Find the component of the resultant of the three forces
    (a) in the \(x\)-direction,
    (b) in the \(y\)-direction.
  2. Hence find the direction of the resultant. \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-3_414_833_267_657} A block of mass 8 kg is at rest on a plane inclined at \(20 ^ { \circ }\) to the horizontal. The block is connected to a vertical wall at the top of the plane by a string. The string is taut and parallel to a line of greatest slope of the plane (see diagram).
  3. Given that the tension in the string is 13 N , find the frictional and normal components of the force exerted on the block by the plane. The string is cut; the block remains at rest, but is on the point of slipping down the plane.
  4. Find the coefficient of friction between the block and the plane.
CAIE M1 2009 June Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-3_165_1417_1320_365} A cyclist and his machine have a total mass of 80 kg . The cyclist starts from rest at the top \(A\) of a straight path \(A B\), and freewheels (moves without pedalling or braking) down the path to \(B\). The path \(A B\) is inclined at \(2.6 ^ { \circ }\) to the horizontal and is of length 250 m (see diagram).
  1. Given that the cyclist passes through \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\), find the gain in kinetic energy and the loss in potential energy of the cyclist and his machine. Hence find the work done against the resistance to motion of the cyclist and his machine. The cyclist continues to freewheel along a horizontal straight path \(B D\) until he reaches the point \(C\), where the distance \(B C\) is \(d \mathrm {~m}\). His speed at \(C\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant, and is the same on \(B D\) as on \(A B\).
  2. Find the value of \(d\). The cyclist starts to pedal at \(C\), generating 425 W of power.
  3. Find the acceleration of the cyclist immediately after passing through \(C\).
CAIE M1 2009 June Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-4_712_526_264_813} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string which passes over a smooth pulley. The system is held at rest with the string taut and its straight parts vertical. Both particles are at a height of 0.36 m above the floor (see diagram). The system is released and \(A\) begins to fall, reaching the floor after 0.6 s .
  1. Find the acceleration of \(A\) as it falls. The mass of \(A\) is 0.45 kg . Find
  2. the tension in the string while \(A\) is falling,
  3. the mass of \(B\),
  4. the maximum height above the floor reached by \(B\).
CAIE M1 2009 June Q7
14 marks Moderate -0.8
7 A particle \(P\) travels in a straight line from \(A\) to \(D\), passing through the points \(B\) and \(C\). For the section \(A B\) the velocity of the particle is \(\left( 0.5 t - 0.01 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(t \mathrm {~s}\) is the time after leaving \(A\).
  1. Given that the acceleration of \(P\) at \(B\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the time taken for \(P\) to travel from \(A\) to \(B\). The acceleration of \(P\) from \(B\) to \(C\) is constant and equal to \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that \(P\) reaches \(C\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time taken for \(P\) to travel from \(B\) to \(C\). \(P\) travels with constant deceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from \(C\) to \(D\). Given that the distance \(C D\) is 300 m , find
  3. the speed with which \(P\) reaches \(D\),
  4. the distance \(A D\). \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 Q1
4 marks Standard +0.3
1 A car of mass 1150 kg travels up a straight hill inclined at \(1.2 ^ { \circ }\) to the horizontal. The resistance to motion of the car is 975 N . Find the acceleration of the car at an instant when it is moving with speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at a power of 35 kW .
CAIE M1 2010 June Q2
5 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-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.