Questions — CAIE M1 (732 questions)

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CAIE M1 2004 June Q6
6 A car of mass 1200 kg travels along a horizontal straight road. The power of the car's engine is 20 kW . The resistance to the car's motion is 400 N .
  1. Find the speed of the car at an instant when its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Show that the maximum possible speed of the car is \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done by the car's engine as the car travels from a point \(A\) to a point \(B\) is 1500 kJ .
  3. Given that the car is travelling at its maximum possible speed between \(A\) and \(B\), find the time taken to travel from \(A\) to \(B\).
CAIE M1 2004 June Q7
7 A particle \(P _ { 1 }\) is projected vertically upwards, from horizontal ground, with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant another particle \(P _ { 2 }\) is projected vertically upwards from the top of a tower of height 25 m , with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the time for which \(P _ { 1 }\) is higher than the top of the tower,
  2. the velocities of the particles at the instant when the particles are at the same height,
  3. the time for which \(P _ { 1 }\) is higher than \(P _ { 2 }\) and is moving upwards.
CAIE M1 2005 June Q1
1 A small block is pulled along a rough horizontal floor at a constant speed of \(1.5 \mathrm {~ms} ^ { - 1 }\) by a constant force of magnitude 30 N acting at an angle of \(\theta ^ { \circ }\) upwards from the horizontal. Given that the work done by the force in 20 s is 720 J , calculate the value of \(\theta\).
CAIE M1 2005 June Q2
6 marks
2
\includegraphics[max width=\textwidth, alt={}, center]{9bb53600-e7ba-4228-84ae-d8ddf7649387-2_350_688_493_731} Three coplanar forces act at a point. The magnitudes of the forces are \(5 \mathrm {~N} , 6 \mathrm {~N}\) and 7 N , and the directions in which the forces act are shown in the diagram. Find the magnitude and direction of the resultant of the three forces.
\(3 A\) and \(B\) are points on the same line of greatest slope of a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. \(A\) is higher up the plane than \(B\) and the distance \(A B\) is 2.25 m . A particle \(P\), of mass \(m \mathrm {~kg}\), is released from rest at \(A\) and reaches \(B 1.5\) s later. Find the coefficient of friction between \(P\) and the plane. [6]
CAIE M1 2005 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{9bb53600-e7ba-4228-84ae-d8ddf7649387-2_478_597_1398_776} Particles \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. Particle \(A\) hangs freely and particle \(B\) is in contact with the table (see diagram).
  1. The system is in limiting equilibrium with the string taut and \(A\) about to move downwards. Find the coefficient of friction between \(B\) and the table. A force now acts on particle \(B\). This force has a vertical component of 1.8 N upwards and a horizontal component of \(X\) N directed away from the pulley.
  2. The system is now in limiting equilibrium with the string taut and \(A\) about to move upwards. Find \(X\).
CAIE M1 2005 June Q5
5 A particle \(P\) moves along the \(x\)-axis in the positive direction. The velocity of \(P\) at time \(t \mathrm {~s}\) is \(0.03 t ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 5\) the displacement of \(P\) from the origin \(O\) is 2.5 m .
  1. Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\).
  2. Find the velocity of \(P\) when its displacement from \(O\) is 11.25 m .
CAIE M1 2005 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{9bb53600-e7ba-4228-84ae-d8ddf7649387-3_735_1484_625_333} The diagram shows the velocity-time graph for a lift moving between floors in a building. The graph consists of straight line segments. In the first stage the lift travels downwards from the ground floor for 5 s , coming to rest at the basement after travelling 10 m .
  1. Find the greatest speed reached during this stage. The second stage consists of a 10 s wait at the basement. In the third stage, the lift travels upwards until it comes to rest at a floor 34.5 m above the basement, arriving 24.5 s after the start of the first stage. The lift accelerates at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 3 s of the third stage, reaching a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  2. the value of \(V\),
  3. the time during the third stage for which the lift is moving at constant speed,
  4. the deceleration of the lift in the final part of the third stage.
CAIE M1 2005 June Q7
7 A car of mass 1200 kg travels along a horizontal straight road. The power provided by the car's engine is constant and equal to 20 kW . The resistance to the car's motion is constant and equal to 500 N . The car passes through the points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The car takes 30.5 s to travel from \(A\) to \(B\).
  1. Find the acceleration of the car at \(A\).
  2. By considering work and energy, find the distance \(A B\).
CAIE M1 2006 June Q1
1 A car of mass 1200 kg travels on a horizontal straight road with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Given that the car's speed increases from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while travelling a distance of 525 m , find the value of \(a\). The car's engine exerts a constant driving force of 900 N . The resistance to motion of the car is constant and equal to \(R \mathrm {~N}\).
  2. Find \(R\).
CAIE M1 2006 June Q2
2 A motorcyclist starts from rest at \(A\) and travels in a straight line until he comes to rest again at \(B\). The velocity of the motorcyclist \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t - 0.01 t ^ { 2 }\). Find
  1. the time taken for the motorcyclist to travel from \(A\) to \(B\),
  2. the distance \(A B\).
CAIE M1 2006 June Q3
6 marks
3
\includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-2_508_1011_1096_568} A particle \(P\) is in equilibrium on a smooth horizontal table under the action of horizontal forces of magnitudes \(F\) N, \(F\) N, \(G\) N and 12 N acting in the directions shown. Find the values of \(F\) and \(G\). [6]
CAIE M1 2006 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_568_1084_269_532} The diagram shows the velocity-time graph for the motion of a small stone which falls vertically from rest at a point \(A\) above the surface of liquid in a container. The downward velocity of the stone \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The stone hits the surface of the liquid with velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0.7\). It reaches the bottom of the container with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 1.2\).
  1. Find
    (a) the height of \(A\) above the surface of the liquid,
    (b) the depth of liquid in the container.
  2. Find the deceleration of the stone while it is moving in the liquid.
  3. Given that the resistance to motion of the stone while it is moving in the liquid has magnitude 0.7 N , find the mass of the stone.
CAIE M1 2006 June Q5
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
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
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
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
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
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
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
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
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
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
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
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
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 \(F\).