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CAIE M1 2023 June Q7
13 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-10_551_776_260_689} Two particles \(P\) and \(Q\), of masses 2 kg and 0.25 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(P\) is on an inclined plane at an angle of \(30 ^ { \circ }\) to the horizontal. Particle \(Q\) hangs below the pulley. Three points \(A , B\) and \(C\) lie on a line of greatest slope of the plane with \(A B = 0.8 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram). Particle \(P\) is released from rest at \(A\) with the string taut and slides down the plane. During the motion of \(P\) from \(A\) to \(C , Q\) does not reach the pulley. The part of the plane from \(A\) to \(B\) is rough, with coefficient of friction 0.3 between the plane and \(P\). The part of the plane from \(B\) to \(C\) is smooth.
    1. Find the acceleration of \(P\) between \(A\) and \(B\).
    2. Hence, find the speed of \(P\) at \(C\).
  2. Find the time taken for \(P\) to travel from \(A\) to \(C\).
    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 2023 June Q1
3 marks Standard +0.3
1 Two particles \(P\) and \(Q\), of masses 0.1 kg and 0.4 kg respectively, are free to move on a smooth horizontal plane. Particle \(P\) is projected with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After \(P\) and \(Q\) collide, the speeds of \(P\) and \(Q\) are equal. Find the two possible values of the speed of \(P\) after the collision.
CAIE M1 2023 June Q2
4 marks Moderate -0.3
2 A car of mass 1500 kg is towing a trailer of mass \(m \mathrm {~kg}\) along a straight horizontal road. The car and the trailer are connected by a tow-bar which is horizontal, light and rigid. There is a resistance force of \(F \mathrm {~N}\) on the car and a resistance force of 200 N on the trailer. The driving force of the car's engine is 3200 N , the acceleration of the car is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and the tension in the tow-bar is 300 N . Find the value of \(m\) and the value of \(F\).
CAIE M1 2023 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-04_337_661_262_740} A smooth ring \(R\) of mass 0.2 kg is threaded on a light string \(A R B\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(A B R = 90 ^ { \circ }\). The angle between the part \(A R\) of the string and the vertical is \(60 ^ { \circ }\). The ring is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), acting on the ring in a direction perpendicular to \(A R\) (see diagram). Calculate the tension in the string and the value of \(X\).
CAIE M1 2023 June Q4
7 marks Standard +0.3
4 A lorry of mass 15000 kg moves on a straight horizontal road in the direction from \(A\) to \(B\). It passes \(A\) and \(B\) with speeds \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The power of the lorry's engine is constant and there is a constant resistance to motion of magnitude 6000 N . The acceleration of the lorry at \(B\) is 0.5 times the acceleration of the lorry at \(A\).
  1. Show that the power of the lorry's engine is 200 kW , and hence find the acceleration of the lorry when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The lorry begins to ascend a straight hill inclined at \(1 ^ { \circ }\) to the horizontal. It is given that the power of the lorry's engine and the resistance force do not change.
  2. Find the steady speed up the hill that the lorry could maintain.
CAIE M1 2023 June Q5
10 marks Standard +0.3
5 A particle starts from rest from a point \(O\) and moves in a straight line. The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = k t ^ { \frac { 1 } { 2 } }\) for \(0 \leqslant t \leqslant 9\) and where \(k\) is a constant. The velocity of the particle at \(t = 9\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 0.1\).
    For \(t > 9\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle is given by \(v = 0.2 ( t - 9 ) ^ { 2 } + 1.8\).
  2. Show that the distance travelled in the first 9 seconds is one tenth of the distance travelled between \(t = 9\) and \(t = 18\).
  3. Find the greatest acceleration of the particle during the first 10 seconds of its motion.
CAIE M1 2023 June Q6
11 marks Standard +0.3
6 An elevator is pulled vertically upwards by a cable. The elevator accelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s , then travels at constant speed for 25 s . The elevator then decelerates at \(0.2 \mathrm {~ms} ^ { - 2 }\) until it comes to rest.
  1. Find the greatest speed of the elevator and hence draw a velocity-time graph for the motion of the elevator.
  2. Find the total distance travelled by the elevator.
    The mass of the elevator is 1200 kg and there is a crate of mass \(m \mathrm {~kg}\) resting on the floor of the elevator.
  3. Given that the tension in the cable when the elevator is decelerating is 12250 N , find the value of \(m\).
  4. Find the greatest magnitude of the force exerted on the crate by the floor of the elevator, and state its direction.
CAIE M1 2023 June Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-10_525_885_264_625} The diagram shows the vertical cross-section \(X Y Z\) of a rough slide. The section \(Y Z\) is a straight line of length 2 m inclined at an angle of \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The section \(Y Z\) is tangential to the curved section \(X Y\) at \(Y\), and \(X\) is 1.8 m above the level of \(Y\). A child of mass 25 kg slides down the slide, starting from rest at \(X\). The work done by the child against the resistance force in moving from \(X\) to \(Y\) is 50 J .
  1. Find the speed of the child at \(Y\).
    It is given that the child comes to rest at \(Z\).
  2. Use an energy method to find the coefficient of friction between the child and \(Y Z\), giving your answer as a fraction in its simplest form.
    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 2024 June Q1
4 marks Easy -1.3
1 A car starts from rest and accelerates at \(2 \mathrm {~ms} ^ { - 2 }\) for 10 s . It then travels at a constant speed for 30 s . The car then uniformly decelerates to rest over a period of 20 s .
  1. Sketch a velocity-time graph for the motion of the car. \includegraphics[max width=\textwidth, alt={}, center]{2af7fd9a-aa78-4d77-aa4e-c01604c8b0ae-03_762_1081_447_493}
  2. Find the total distance travelled by the car.
CAIE M1 2024 June Q2
7 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{2af7fd9a-aa78-4d77-aa4e-c01604c8b0ae-04_558_606_276_715} Two forces of magnitudes 20 N and \(F \mathrm {~N}\) act at a point \(P\) in the directions shown in the diagram.
  1. Given that the resultant force has no component in the \(y\)-direction, calculate the value of \(F\).
  2. Given instead that \(F = 10\), find the magnitude and direction of the resultant force.
CAIE M1 2024 June Q3
4 marks Standard +0.3
3 A train of mass 180000 kg ascends a straight hill of length 1.5 km , inclined at an angle of \(1.5 ^ { \circ }\) to the horizontal. As it ascends the hill, the total work done to overcome the resistance to motion is 12000 kJ and the speed of the train decreases from \(45 \mathrm {~ms} ^ { - 1 }\) to \(40 \mathrm {~ms} ^ { - 1 }\). Find the work done by the engine of the train as it ascends the hill, giving your answer in kJ .
CAIE M1 2024 June Q4
6 marks Moderate -0.3
4 A car of mass 1700 kg is pulling a trailer of mass 300 kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of 400 N on the car and 150 N on the trailer. The power of the car's engine is 14000 W . Find the acceleration of the car and the tension in the cable when the speed is \(20 \mathrm {~ms} ^ { - 1 }\).
CAIE M1 2024 June Q5
8 marks Moderate -0.3
5 A straight slope of length 60 m is inclined at an angle of \(12 ^ { \circ }\) to the horizontal. A bobsled starts at the top of the slope with a speed of \(5 \mathrm {~ms} ^ { - 1 }\). The bobsled slides directly down the slope.
  1. It is given that there is no resistance to the bobsled's motion. Find its speed when it reaches the bottom of the slope.
  2. It is given instead that the coefficient of friction between the bobsled and the slope is 0.03 . Find the time that it takes for the bobsled to reach the bottom of the slope.
CAIE M1 2024 June Q6
11 marks Standard +0.3
6 A particle moves in a straight line, starting from a point \(O\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v \mathrm {~ms} ^ { - 1 }\). It is given that \(\mathbf { v } = \mathrm { kt } ^ { \frac { 1 } { 2 } } - 2 \mathrm { t } - 8\), where \(k\) is a positive constant. The maximum velocity of the particle is \(4.5 \mathrm {~ms} ^ { - 1 }\).
  1. Show that \(k = 10\).
    1. Verify that \(v = 0\) when \(t = 1\) and \(t = 16\).
    2. Find the distance travelled by the particle in the first 16 s .
CAIE M1 2024 June Q7
10 marks Standard +0.8
7 A particle \(P\) of mass 0.2 kg is projected vertically upwards from horizontal ground with speed \(25 \mathrm {~ms} ^ { - 1 }\).
  1. Show that the speed of \(P\) when it reaches 20 m above the ground is \(15 \mathrm {~ms} ^ { - 1 }\).
    When \(P\) reaches 20 m above the ground it collides with a second particle \(Q\) of mass 0.1 kg which is moving downwards at \(20 \mathrm {~ms} ^ { - 1 } . P\) is brought to instantaneous rest in the collision.
  2. Find the velocity of \(Q\) immediately after the collision.
    When \(P\) reaches the ground it rebounds back directly upwards with half of the speed that it had immediately before hitting the ground.
  3. Find the height above the ground at which \(P\) and \(Q\) next collide.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE M1 2024 June Q1
3 marks Moderate -0.3
1 A cyclist and bicycle have a total mass of 72 kg . The cyclist rides along a horizontal road against a total resistance force of 28 N . Find the total work done by the cyclist to increase his speed from \(8 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\) while travelling a distance of 100 metres.
CAIE M1 2024 June Q2
5 marks Standard +0.3
2 A particle \(P\) moves in a straight line. At time \(t\) s after leaving a point \(O\) on the line, \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\), where \(\mathrm { v } = 44 \mathrm { t } - 6 \mathrm { t } ^ { 2 } - 36\).
  1. Find the set of values of \(t\) for which the acceleration of the particle is positive.
  2. Find the two values of \(t\) at which \(P\) returns to \(O\). \includegraphics[max width=\textwidth, alt={}, center]{3eaf3652-ff91-4bae-9f20-83487d635612-04_714_796_248_635} Four coplanar forces of magnitude \(P \mathrm {~N} , 10 \mathrm {~N} , 16 \mathrm {~N}\) and 2 N act at a point in the directions shown in the diagram. It is given that the forces are in equilibrium. Find the values of \(\theta\) and \(P\).
CAIE M1 2024 June Q4
7 marks Standard +0.3
4 A car has mass 1400 kg . When the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\) the magnitude of the resistance to motion is \(\mathrm { kv } ^ { 2 } \mathrm {~N}\) where \(k\) is a constant.
  1. The car moves at a constant speed of \(24 \mathrm {~ms} ^ { - 1 }\) up a hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.12\). At this speed the magnitude of the resistance to motion is 480 N .
    1. Find the value of \(k\).
    2. Find the power of the car's engine.
  2. The car now moves at a constant speed on a straight level road. Given that its engine is working at 54 kW , find this speed. \includegraphics[max width=\textwidth, alt={}, center]{3eaf3652-ff91-4bae-9f20-83487d635612-06_542_923_251_571} A particle of mass 0.8 kg lies on a rough plane which is inclined at an angle of \(28 ^ { \circ }\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(T \mathrm {~N}\). This force acts at an angle of \(35 ^ { \circ }\) above a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.2 . Find the least and greatest possible values of \(T\).
CAIE M1 2024 June Q6
11 marks Standard +0.3
6 Three particles \(A , B\) and \(C\) of masses \(5 \mathrm {~kg} , 1 \mathrm {~kg}\) and 2 kg respectively lie at rest in that order on a straight smooth horizontal track \(X Y Z\). Initially \(A\) is at \(X , B\) is at \(Y\) and \(C\) is at \(Z\). Particle \(A\) is projected towards \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and at the same instant \(C\) is projected towards \(B\) with a speed of \(v \mathrm {~ms} ^ { - 1 }\). In the subsequent motion, \(A\) collides and coalesces with \(B\) to form particle \(D\). Particle \(D\) then collides and coalesces with \(C\) to form particle \(E\) and \(E\) moves towards \(Z\).
  1. Show that after the second collision the speed of \(E\) is \(\frac { 15 - v } { 4 } \mathrm {~ms} ^ { - 1 }\).
  2. The total loss of kinetic energy of the system due to the two collisions is 63 J . Use the result from (a) to show that \(v = 3\).
  3. It is given that the distance \(X Y\) is 36 m and the distance \(Y Z\) is 98 m .
    1. Find the time between the two collisions.
    2. Find the time between the instant that \(A\) is projected from \(X\) and the instant that \(E\) reaches \(Z\).
CAIE M1 2024 June Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{3eaf3652-ff91-4bae-9f20-83487d635612-10_621_908_248_580} Two particles \(P\) and \(Q\) of masses 2.5 kg and 0.5 kg respectively are connected by a light inextensible string that passes over a small smooth pulley fixed at the top of a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Particle \(P\) is on the plane and \(Q\) hangs below the pulley such that the level of \(Q\) is 2 m below the level of \(P\) (see diagram). Particle \(P\) is released from rest with the string taut and slides down the plane. The plane is rough with coefficient of friction 0.2 between the plane and \(P\).
  1. Find the acceleration of \(P\).
  2. Use an energy method to find the speed of the particles at the instant when they are at the same vertical height.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE M1 2020 March Q1
4 marks Moderate -0.3
1 A lorry of mass 16000 kg is travelling along a straight horizontal road. The engine of the lorry is working at constant power. The work done by the driving force in 10 s is 750000 J .
  1. Find the power of the lorry's engine.
  2. There is a constant resistance force acting on the lorry of magnitude 2400 N . Find the acceleration of the lorry at an instant when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2020 March Q2
6 marks Moderate -0.5
2 A particle \(P\) of mass 0.4 kg is on a rough horizontal floor. The coefficient of friction between \(P\) and the floor is \(\mu\). A force of magnitude 3 N is applied to \(P\) upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The particle is initially at rest and accelerates at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the time it takes for \(P\) to travel a distance of 1.44 m from its starting point.
  2. Find \(\mu\).
CAIE M1 2020 March Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-04_519_1018_260_561} The diagram shows the vertical cross-section of a surface. \(A , B\) and \(C\) are three points on the crosssection. The level of \(B\) is \(h \mathrm {~m}\) above the level of \(A\). The level of \(C\) is 0.5 m below the level of \(A\). A particle of mass 0.2 kg is projected up the slope from \(A\) with initial speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle remains in contact with the surface as it travels from \(A\) to \(C\).
  1. Given that the particle reaches \(B\) with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there is no resistance force, find \(h\).
  2. It is given instead that there is a resistance force and that the particle does 3.1 J of work against the resistance force as it travels from \(A\) to \(C\). Find the speed of the particle when it reaches \(C\).
CAIE M1 2020 March Q4
7 marks Standard +0.3
4 A cyclist travels along a straight road with constant acceleration. He passes through points \(A , B\) and \(C\). The cyclist takes 2 seconds to travel along each of the sections \(A B\) and \(B C\) and passes through \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is \(\frac { 4 } { 5 }\) of the distance \(B C\).
  1. Find the acceleration of the cyclist.
  2. Find \(A C\).
CAIE M1 2020 March Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-08_572_572_262_790} Coplanar forces, of magnitudes \(F \mathrm {~N} , 3 \mathrm {~N} , 6 \mathrm {~N}\) and 4 N , act at a point \(P\), as shown in the diagram.
  1. Given that \(\alpha = 60\), and that the resultant of the four forces is in the direction of the 3 N force, find \(F\).
  2. Given instead that the four forces are in equilibrium, find the values of \(F\) and \(\alpha\).