Questions — CAIE M1 (786 questions)

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CAIE M1 2022 November Q1
3 marks Easy -1.2
1 A particle \(P\) is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .
  1. Find \(u\).
  2. Find the greatest height of \(P\) above the ground.
CAIE M1 2022 November Q2
4 marks Moderate -0.3
2 A box of mass 5 kg is pulled at a constant speed of \(1.8 \mathrm {~ms} ^ { - 1 }\) for 15 s up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves along a line of greatest slope against a frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope.
  1. Find the change in gravitational potential energy of the box.
  2. Find the work done by the pulling force.
CAIE M1 2022 November Q3
6 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-05_610_591_257_778} A ring of mass 4 kg is threaded on a smooth circular rigid wire with centre \(C\). The wire is fixed in a vertical plane and the ring is kept at rest by a light string connected to \(A\), the highest point of the circle. The string makes an angle of \(25 ^ { \circ }\) to the vertical (see diagram). Find the tension in the string and the magnitude of the normal reaction of the wire on the ring.
CAIE M1 2022 November Q4
7 marks Standard +0.3
4 A particle \(P\) travels in the positive direction along a straight line with constant acceleration. \(P\) travels a distance of 52 m during the 2 nd second of its motion and a distance of 64 m during the 4th second of its motion.
  1. Find the initial speed and the acceleration of \(P\).
  2. Find the distance travelled by \(P\) during the first 10 seconds of its motion.
CAIE M1 2022 November Q5
8 marks Standard +0.8
5 Particles \(X\) and \(Y\) move in a straight line through points \(A\) and \(B\). Particle \(X\) starts from rest at \(A\) and moves towards \(B\). At the same instant, \(Y\) starts from rest at \(B\). At time \(t\) seconds after the particles start moving
  • the acceleration of \(X\) in the direction \(A B\) is given by \(( 12 t + 12 ) \mathrm { m } \mathrm { s } ^ { - 2 }\),
  • the acceleration of \(Y\) in the direction \(A B\) is given by \(( 24 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
    1. It is given that the velocities of \(X\) and \(Y\) are equal when they collide.
Calculate the distance \(A B\).
  • It is given instead that \(A B = 36 \mathrm {~m}\). Verify that \(X\) and \(Y\) collide after 3 s.
    •
    CAIE M1 2022 November Q6
    10 marks Standard +0.3
    6 A car of mass 1750 kg is pulling a caravan of mass 500 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 650 N and 150 N respectively.
    1. The car and caravan are moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
      1. Find the power of the car's engine.
      2. The engine's power is now suddenly increased to 40 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
    2. The car and caravan now travel up a straight hill, inclined at an angle \(\sin ^ { - 1 } 0.14\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 31 kW . The resistances to the motion of the car and caravan are unchanged. Find \(v\).
    CAIE M1 2022 November Q7
    12 marks Challenging +1.2
    7 \includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-12_560_716_258_712} Particles of masses 1.5 kg and 3 kg lie on a plane which is inclined at an angle of \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The section of the plane from \(A\) to \(B\) is smooth and the section of the plane from \(B\) to \(C\) is rough. The 1.5 kg particle is held at rest at \(A\) and the 3 kg particle is in limiting equilibrium at \(B\). The distance \(A B\) is \(x \mathrm {~m}\) and the distance \(B C\) is 4 m (see diagram).
    1. Show that the coefficient of friction between the particle at \(B\) and the plane is 0.75 .
      The 1.5 kg particle is released from rest. In the subsequent motion the two particles collide and coalesce. The time taken for the combined particle to travel from \(B\) to \(C\) is 2 s . The coefficient of friction between the combined particle and the plane is still 0.75 .
    2. Find \(x\).
    3. Find the total loss of energy of the particles from the time the 1.5 kg particle is released until the combined particle reaches \(C\).
      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 November Q1
    3 marks Standard +0.3
    1 A particle of mass 1.6 kg is projected with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a smooth plane inclined at \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.
    CAIE M1 2023 November Q2
    5 marks Moderate -0.5
    2 \includegraphics[max width=\textwidth, alt={}, center]{f1f33ef0-0d4d-4a4a-aadb-28de8dc0ea8d-03_280_588_264_774} A particle of mass 2.4 kg is held in equilibrium by two light inextensible strings, one of which is attached to point \(A\) and the other attached to point \(B\). The strings make angles of \(35 ^ { \circ }\) and \(40 ^ { \circ }\) with the horizontal (see diagram). Find the tension in each of the two strings.
    CAIE M1 2023 November Q3
    8 marks Moderate -0.8
    3 \includegraphics[max width=\textwidth, alt={}, center]{f1f33ef0-0d4d-4a4a-aadb-28de8dc0ea8d-04_666_1278_280_424} The diagram shows the velocity-time graph for the motion of a bus. The bus starts from rest and accelerates uniformly for 8 seconds until it reaches a speed of \(12.6 \mathrm {~ms} ^ { - 1 }\). The bus maintains this speed for 40 seconds. It then decelerates uniformly in two stages. Between 48 and 62 seconds the bus decelerates at \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and between 62 and 70 seconds it decelerates at \(2 a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until coming to rest.
    1. Find the distance covered by the bus in the first 8 seconds.
    2. Find the value of \(a\).
    3. Find the average speed of the bus for the whole journey.
    CAIE M1 2023 November Q4
    9 marks Standard +0.3
    4 Two particles \(P\) and \(Q\), of masses 6 kg and 2 kg respectively, lie at rest 12.5 m apart on a rough horizontal plane. The coefficient of friction between each particle and the plane is 0.4 . Particle \(P\) is projected towards \(Q\) with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that the speed of \(P\) immediately before the collision with \(Q\) is \(10 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
      In the collision \(P\) and \(Q\) coalesce to form particle \(R\).
    2. Find the loss of kinetic energy due to the collision.
      The coefficient of friction between \(R\) and the plane is 0.4 .
    3. Find the distance travelled by particle \(R\) before coming to rest.
    CAIE M1 2023 November Q5
    7 marks Standard +0.3
    5 \includegraphics[max width=\textwidth, alt={}, center]{f1f33ef0-0d4d-4a4a-aadb-28de8dc0ea8d-08_483_840_258_649} The diagram shows a particle \(A\), of mass 1.2 kg , which lies on a plane inclined at an angle of \(40 ^ { \circ }\) to the horizontal and a particle \(B\), of mass 1.6 kg , which lies on a plane inclined at an angle of \(50 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the planes. The parts \(A P\) and \(B P\) of the string are taut and parallel to lines of greatest slope of the respective planes. The two planes are rough, with the same coefficient of friction, \(\mu\), between the particles and the planes. Find the value of \(\mu\) for which the system is in limiting equilibrium.
    CAIE M1 2023 November Q6
    9 marks Moderate -0.3
    6 A car of mass 1300 kg is moving on a straight road.
    1. On a horizontal section of the road, the car has a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and there is a constant force of 650 N resisting the motion.
      1. Calculate, in kW , the power developed by the engine of the car.
      2. Given that this power is suddenly increased by 9 kW , find the instantaneous acceleration of the car.
    2. On a section of the road inclined at \(\sin ^ { - 1 } 0.08\) to the horizontal, the resistance to the motion of the car is \(( 1000 + 20 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels downwards along this section of the road at constant speed with the engine working at 11.5 kW . Find this constant speed.
    CAIE M1 2023 November Q7
    9 marks Challenging +1.2
    7 A particle moves in a straight line starting from a point \(O\) before coming to instantaneous rest at a point \(X\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle is given by $$\begin{array} { l l } v = 7.2 t ^ { 2 } & 0 \leqslant t \leqslant 2 , \\ v = 30.6 - 0.9 t & 2 \leqslant t \leqslant 8 , \\ v = \frac { 1600 } { t ^ { 2 } } + k t & 8 \leqslant t , \end{array}$$ where \(k\) is a constant. It is given that there is no instantaneous change in velocity at \(t = 8\).
    Find the distance \(O X\).
    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 November Q1
    3 marks Moderate -0.3
    1 A particle is projected vertically upwards from horizontal ground with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle has height \(s \mathrm {~m}\) above the ground at times 3 seconds and 4 seconds after projection. Find the value of \(u\) and the value of \(s\).
    CAIE M1 2023 November Q2
    5 marks Moderate -0.5
    2 \includegraphics[max width=\textwidth, alt={}, center]{99f20949-471d-4da3-a680-ec24abf6baa5-03_510_604_260_769} A machine for driving a nail into a block of wood causes a hammerhead to drop vertically onto the top of a nail. The mass of the hammerhead is 1.2 kg and the mass of the nail is 0.004 kg (see diagram). The hammerhead hits the nail with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and remains in contact with the nail after the impact. The combined hammerhead and nail move immediately after the impact with speed \(40 \mathrm {~ms} ^ { - 1 }\).
    1. Calculate \(v\), giving your answer as an exact fraction.
    2. The nail is driven 4 cm into the wood. Find the constant force resisting the motion.
    CAIE M1 2023 November Q3
    6 marks Moderate -0.8
    3 A block of mass 8 kg slides down a rough plane inclined at \(30 ^ { \circ }\) to the horizontal, starting from rest. The coefficient of friction between the block and the plane is \(\mu\). The block accelerates uniformly down the plane at \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Draw a diagram showing the forces acting on the block.
    2. Find the value of \(\mu\).
    3. Find the speed of the block after it has moved 3 m down the plane.
    CAIE M1 2023 November Q4
    7 marks Standard +0.3
    4 A car has mass 1600 kg .
    1. The car is moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is subject to a constant resistance of magnitude 480 N . Find, in kW , the rate at which the engine of the car is working.
      The car now moves down a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.09\). The engine of the car is working at a constant rate of 12 kW . The speed of the car is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill. Ten seconds later the car has travelled 280 m down the hill and has speed \(32 \mathrm {~ms} ^ { - 1 }\).
    2. Given that the resistance is not constant, use an energy method to find the total work done against the resistance during the ten seconds.
    CAIE M1 2023 November Q5
    8 marks Moderate -0.8
    5 \includegraphics[max width=\textwidth, alt={}, center]{99f20949-471d-4da3-a680-ec24abf6baa5-06_438_463_264_840} A light string \(A B\) is fixed at \(A\) and has a particle of weight 80 N attached at \(B\). A horizontal force of magnitude \(P \mathrm {~N}\) is applied at \(B\) such that the string makes an angle \(\theta ^ { \circ }\) to the vertical (see diagram).
    1. It is given that \(P = 32\) and the system is in equilibrium. Find the tension in the string and the value of \(\theta\).
    2. It is given instead that the tension in the string is 120 N and that the particle attached at \(B\) still has weight 80 N . Find the value of \(P\) and the value of \(\theta\).
    CAIE M1 2023 November Q6
    8 marks Standard +0.3
    6 A particle moves in a straight line. At time \(t \mathrm {~s}\), the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the particle is given by \(a = 36 - 6 t\). The velocity of the particle is \(27 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 2\).
    1. Find the values of \(t\) when the particle is at instantaneous rest.
    2. Find the total distance the particle travels during the first 12 seconds.
    CAIE M1 2023 November Q7
    13 marks Standard +0.8
    7 \includegraphics[max width=\textwidth, alt={}, center]{99f20949-471d-4da3-a680-ec24abf6baa5-10_335_937_255_605} Particles \(A\) and \(B\), of masses 2.4 kg and 3.3 kg respectively, are connected by a light inextensible string that passes over a smooth pulley which is fixed to the top of a rough plane. The plane makes an angle of \(\theta ^ { \circ }\) with horizontal ground. Particle \(A\) is on the plane and the section of the string between \(A\) and the pulley is parallel to a line of greatest slope of the plane. Particle \(B\) hangs vertically below the pulley and is 1 m above the ground (see diagram). The coefficient of friction between the plane and \(A\) is \(\mu\).
    1. It is given that \(\theta = 30\) and the system is in equilibrium with \(A\) on the point of moving directly up the plane. Show that \(\mu = 1.01\) correct to 3 significant figures.
    2. It is given instead that \(\theta = 20\) and \(\mu = 1.01\). The system is released from rest with the string taut. Find the total distance travelled by \(A\) before coming to instantaneous rest. You may assume that \(A\) does not reach the pulley and that \(B\) remains at rest after it hits 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 2024 November Q2
    4 marks Moderate -0.3
    2 A block of mass 20 kg is held at rest at the top of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The block is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of the plane. There is a resistance force acting on the block. As the block moves 2 m down the plane from its point of projection, the work done against this resistance force is 50 J . Find the speed of the block when it has moved 2 m down the plane. \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-04_2716_38_109_2012}
    CAIE M1 2024 November Q3
    5 marks Standard +0.3
    3 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and his bicycle is 90 kg . The power exerted by the cyclist is 250 W . At an instant when the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).
    1. Find the value of the constant resistance to motion acting on the cyclist.
      The cyclist comes to the bottom of a hill inclined at \(2 ^ { \circ }\) to the horizontal.
    2. Given that the power and resistance to motion are unchanged, find the steady speed which the cyclist could maintain when riding up the hill.
    CAIE M1 2024 November Q4
    6 marks Standard +0.3
    4 \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_389_1134_258_468} The diagram shows two particles, \(A\) and \(B\), of masses 0.2 kg and 0.1 kg respectively. The particles are suspended below a horizontal ceiling by two strings, \(A P\) and \(B Q\), attached to fixed points \(P\) and \(Q\) on the ceiling. The particles are connected by a horizontal string, \(A B\). Angle \(A P Q = 45 ^ { \circ }\) and \(B Q P = \theta ^ { \circ }\). Each string is light and inextensible. The particles are in equilibrium.
    1. Find the value of the tension in the string \(A B\). \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_2715_44_110_2006} \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-07_2721_34_101_20}
    2. Find the value of \(\theta\) and the tension in the string \(B Q\).
    CAIE M1 2024 November Q5
    9 marks Standard +0.8
    5 Two particles, \(P\) and \(Q\), of masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively, are held at rest in the same vertical line. The heights of \(P\) and \(Q\) above horizontal ground are 1 m and 2 m respectively. \(P\) is projected vertically upwards with speed \(2 \mathrm {~ms} ^ { - 1 }\). At the same instant, \(Q\) is released from rest.
    1. Find the speed of each particle immediately before they collide.
    2. It is given that immediately after the collision the downward speed of \(Q\) is \(3.5 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(P\) at the instant that it reaches the ground.