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CAIE M1 2019 November Q4
7 marks Moderate -0.3
4 A lorry of mass 25000 kg travels along a straight horizontal road. There is a constant force of 3000 N resisting the motion.
  1. Find the power required to maintain a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The lorry comes to a straight hill inclined at \(2 ^ { \circ }\) to the horizontal. The driver switches off the engine of the lorry at the point \(A\) which is at the foot of the hill. Point \(B\) is further up the hill. The speeds of the lorry at \(A\) and \(B\) are \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The resistance force is still 3000 N .
  2. Use an energy method to find the height of \(B\) above the level of \(A\).
CAIE M1 2019 November Q5
7 marks Standard +0.3
5 Two particles \(A\) and \(B\) move in the same vertical line. Particle \(A\) is projected vertically upwards from the ground with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). One second later particle \(B\) is dropped from rest from a height of 40 m .
  1. Find the height above the ground at which the two particles collide.
  2. Find the difference in the speeds of the two particles at the instant when the collision occurs.
CAIE M1 2019 November Q6
11 marks Standard +0.8
6 A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac { 24 } { 25 }\). The force is applied for a period of 5 s , during which time the block moves a distance of 4.5 m .
  1. Find the magnitude of the frictional force on the block.
  2. Show that the coefficient of friction between the block and the plane is 0.165 , correct to 3 significant figures.
  3. When the block has moved a distance of 4.5 m , the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion. \includegraphics[max width=\textwidth, alt={}, center]{cb1cc219-608f-4f11-ab2c-97cc8f0798c7-12_512_1097_258_523} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a smooth plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 } . P\) lies on the plane and \(Q\) hangs vertically below the pulley at a height of 0.8 m above the floor (see diagram). The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. \(P\) is released from rest and \(Q\) moves vertically downwards.
  4. Find the tension in the string and the magnitude of the acceleration of the particles. \(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.
  5. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor.
  6. When \(Q\) hits the floor the string becomes slack. Find the time, from the instant at which \(P\) is released, for the string to become taut again.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2019 November Q1
3 marks Easy -1.2
1 A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pulling force of 2500 N and the crate is moving at constant speed. Find the coefficient of friction between the crate and the ground.
CAIE M1 2019 November Q2
5 marks Standard +0.3
2 A train of mass 150000 kg ascends a straight slope inclined at \(\alpha ^ { \circ }\) to the horizontal with a constant driving force of 16000 N . At a point \(A\) on the slope the speed of the train is \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Point \(B\) on the slope is 500 m beyond \(A\). At \(B\) the speed of the train is \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a resistance force acting on the train and the train does \(4 \times 10 ^ { 6 } \mathrm {~J}\) of work against this resistance force between \(A\) and \(B\). Find the value of \(\alpha\).
CAIE M1 2019 November Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-05_479_647_264_749} Three coplanar forces of magnitudes \(50 \mathrm {~N} , 60 \mathrm {~N}\) and 100 N act at a point. The resultant of the forces has magnitude \(R \mathrm {~N}\). The directions of these forces are shown in the diagram. Find the values of \(R\) and \(\alpha\).
CAIE M1 2019 November Q4
6 marks Standard +0.3
4 A car travels along a straight road with constant acceleration. It passes through points \(P , Q , R\) and \(S\). The times taken for the car to travel from \(P\) to \(Q , Q\) to \(R\) and \(R\) to \(S\) are each equal to 10 s . The distance \(Q R\) is 1.5 times the distance \(P Q\). At point \(Q\) the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the acceleration of the car is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the distance \(Q S\) and hence find the average speed of the car between \(Q\) and \(S\).
CAIE M1 2019 November Q5
8 marks Moderate -0.3
5 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and his bicycle is 80 kg . His power output is a constant 240 W . His acceleration when he is travelling at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that the resistance to the cyclist's motion is 16 N .
  2. Find the steady speed that the cyclist can maintain if his power output and the resistance force are both unchanged.
  3. The cyclist later ascends a straight hill inclined at \(3 ^ { \circ }\) to the horizontal. His power output and the resistance force are still both unchanged. Find his acceleration when he is travelling at \(4 \mathrm {~ms} ^ { - 1 }\).
CAIE M1 2019 November Q6
9 marks Standard +0.3
6 Particle \(P\) travels in a straight line from \(A\) to \(B\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.04 t ^ { 3 } + c t ^ { 2 } + k t$$ \(P\) takes 5 s to travel from \(A\) to \(B\) and it reaches \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is 25 m .
  1. Find the values of the constants \(c\) and \(k\).
  2. Show that the acceleration of \(P\) is a minimum when \(t = 2.5\).
CAIE M1 2019 November Q7
13 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-12_565_511_260_817} Two particles \(A\) and \(B\) have masses \(m \mathrm {~kg}\) and \(k m \mathrm {~kg}\) respectively, where \(k > 1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.81 m above horizontal ground (see diagram). The system is released from rest and particle \(B\) reaches the ground 0.9 s later. The particle \(A\) does not reach the pulley in its subsequent motion.
  1. Find the value of \(k\) and show that the tension in the string before \(B\) reaches the ground is equal to \(12 m \mathrm {~N}\).
    At the instant when \(B\) reaches the ground, the string breaks.
  2. Show that the speed of \(A\) when it reaches the ground is \(5.97 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures, and find the time taken, after the string breaks, for \(A\) to reach the ground.
  3. Sketch a velocity-time graph for the motion of particle \(A\) from the instant when the system is released until \(A\) reaches the ground. If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 Specimen Q1
4 marks Easy -1.2
1 A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.
  1. Find the work done by the weightlifter.
  2. Given that the time taken to raise the mass is 1.2 s , find the average power developed by the weightlifter.
CAIE M1 Specimen Q2
6 marks Moderate -0.3
2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
  1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
  2. Find the distance that the particle travels along the ground before it comes to rest.
CAIE M1 Specimen Q3
6 marks Moderate -0.3
3 A lorry of mass 24000 kg is travelling up a hill which is inclined at \(3 ^ { \circ }\) to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N .
  1. When the speed of the lorry is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power developed by the lorry's engine.
  2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N . \includegraphics[max width=\textwidth, alt={}, center]{75c345bb-7cbd-4b2a-b3a0-0086b80b36c1-05_499_784_258_685} Blocks \(P\) and \(Q\), of mass \(m \mathrm {~kg}\) and 5 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 rough plane inclined at \(35 ^ { \circ }\) to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2 . Find the set of values of \(m\) for which the two blocks remain at rest. \includegraphics[max width=\textwidth, alt={}, center]{75c345bb-7cbd-4b2a-b3a0-0086b80b36c1-06_351_1038_255_557} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(A B\) of length 3 m . Three horizontal forces of magnitudes \(F \mathrm {~N} , 10 \mathrm {~N}\) and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R \mathrm {~N}\) in the direction shown in the diagram.
CAIE M1 Specimen Q6
10 marks Moderate -0.3
6 A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) after leaving \(O\) is given by $$v = 0.6 t - 0.03 t ^ { 2 }$$
  1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time.
  2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity.
CAIE M1 Specimen Q7
10 marks Standard +0.3
7 A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 36 m . The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(A B\) is 210 m .
  1. Find the total time that the cyclist takes to travel from \(A\) to \(B\).
    24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
  2. Find the time that it takes from when the cyclist starts until the car overtakes her.
CAIE M2 2002 June Q1
5 marks Standard +0.3
1 One end of a light elastic string of natural length 1.6 m and modulus of elasticity 25 N is attached to a fixed point \(A\). A particle \(P\) of mass 0.15 kg is attached to the other end of the string. \(P\) is held at rest at a point 2 m vertically below \(A\) and is then released.
  1. For the motion from the instant of release until the string becomes slack, find the loss of elastic potential energy and the gain in gravitational potential energy.
  2. Hence find the speed of \(P\) at the instant the string becomes slack.
CAIE M2 2002 June Q2
4 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_316_1065_712_541} Two identical uniform heavy triangular prisms, each of base width 10 cm , are arranged as shown at the ends of a smooth horizontal shelf of length 1 m . Some books, each of width 5 cm , are placed on the shelf between the prisms.
  1. Find how far the base of a prism can project beyond an end of the shelf without the prism toppling.
  2. Find the greatest number of books that can be stored on the shelf without either of the prisms toppling.
CAIE M2 2002 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_202_972_1619_584} A light elastic string has natural length 0.8 m and modulus of elasticity 12 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level and 0.96 m apart. A particle of weight \(W \mathrm {~N}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.14 m below \(A B\) (see diagram). Find \(W\).
CAIE M2 2002 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_576_826_258_662} A hollow cone with semi-vertical angle \(45 ^ { \circ }\) is fixed with its axis vertical and its vertex \(O\) downwards. A particle \(P\) of mass 0.3 kg moves in a horizontal circle on the inner surface of the cone, which is smooth. \(P\) is attached to one end of a light inextensible string of length 1.2 m . The other end of the string is attached to the cone at \(O\) (see diagram). The string is taut and rotates at a constant angular speed of \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Find the acceleration of \(P\).
  2. Find the tension in the string and the force exerted on \(P\) by the cone.
CAIE M2 2002 June Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_590_754_1425_699} A uniform lamina of weight 9 N has dimensions as shown in the diagram. The lamina is freely hinged to a fixed point at \(A\). A light inextensible string has one end attached to \(B\), and the other end attached to a fixed point \(C\), which is in the same vertical plane as the lamina. The lamina is in equilibrium with \(A B\) horizontal and angle \(A B C = 150 ^ { \circ }\).
  1. Show that the tension in the string is 12.2 N .
  2. Find the magnitude of the force acting on the lamina at \(A\).
CAIE M2 2002 June Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-4_182_844_264_653} A particle \(P\) of mass 0.4 kg travels on a horizontal surface along the line \(O A\) in the direction from \(O\) to \(A\). Air resistance of magnitude \(0.1 v \mathrm {~N}\) opposes the motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\) after it passes through the fixed point \(O\) (see diagram). The speed of \(P\) at \(O\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Assume that the horizontal surface is smooth. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} x } = - \frac { 1 } { 4 }\), where \(x \mathrm {~m}\) is the distance of \(P\) from \(O\) at time \(t \mathrm {~s}\), and hence find the distance from \(O\) at which the speed of \(P\) is zero.
  2. Assume instead that the horizontal surface is not smooth and that the coefficient of friction between \(P\) and the surface is \(\frac { 3 } { 40 }\).
    (a) Show that \(4 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( v + 3 )\).
    (b) Hence find the value of \(t\) for which the speed of \(P\) is zero.
CAIE M2 2002 June Q7
9 marks Standard +0.3
7 A ball is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(T\) s after projection, the ball passes through the point \(A\), whose horizontal and vertically upward displacements from \(O\) are 10 m and 2 m respectively.
  1. By using the equation of the trajectory, or otherwise, find the value of \(V\).
  2. Find the value of \(T\).
  3. Find the angle that the direction of motion of the ball at \(A\) makes with the horizontal.
CAIE M2 2003 June Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_533_497_269_824} A frame consists of a uniform circular ring of radius 25 cm and mass 1.5 kg , and a uniform rod of length 48 cm and mass 0.6 kg . The ends \(A\) and \(B\) of the rod are attached to points on the circumference of the ring, as shown in the diagram. Find the distance of the centre of mass of the frame from the centre of the ring.
CAIE M2 2003 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_439_608_1181_772} A uniform solid hemisphere, with centre \(O\) and radius 4 cm , is held so that a point \(P\) of its rim is in contact with a horizontal surface. The plane face of the hemisphere makes an angle of \(70 ^ { \circ }\) with the horizontal. \(Q\) is the point on the axis of symmetry of the hemisphere which is vertically above \(P\). The diagram shows the vertical cross-section of the hemisphere which contains \(O , P\) and \(Q\).
  1. Determine whether or not the centre of mass of the hemisphere is between \(O\) and \(Q\). The hemisphere is now released.
  2. State whether or not the hemisphere falls on to its plane face, giving a reason for your answer.
CAIE M2 2003 June Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-3_464_439_274_854} A uniform beam \(A B\) has length 6 m and mass 45 kg . One end of a light inextensible rope is attached to the beam at the point 2.5 m from \(A\). The other end of the rope is attached to a fixed point \(P\) on a vertical wall. The beam is in equilibrium with \(A\) in contact with the wall at a point 5 m below \(P\). The rope is taut and at right angles to \(A B\) (see diagram). Find
  1. the tension in the rope,
  2. the horizontal and vertical components of the force exerted by the wall on the beam at \(A\).