Questions M1 (1912 questions)

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CAIE M1 2023 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{308cecda-3bc2-4113-b7dd-ed317c5f32c5-03_638_554_260_792} The diagram shows a smooth ring \(R\), of mass \(m \mathrm {~kg}\), threaded on a light inextensible string. A horizontal force of magnitude 2 N acts on \(R\). The ends of the string are attached to fixed points \(A\) and \(B\) on a vertical wall. The part \(A R\) of the string makes an angle of \(30 ^ { \circ }\) with the vertical, the part \(B R\) makes an angle of \(40 ^ { \circ }\) with the vertical and the string is taut. The ring is in equilibrium. Find the tension in the string and find the value of \(m\).
\includegraphics[max width=\textwidth, alt={}, center]{308cecda-3bc2-4113-b7dd-ed317c5f32c5-04_521_707_259_719} A block of mass 10 kg is at rest on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. A force of 120 N is applied to the block at an angle of \(20 ^ { \circ }\) above a line of greatest slope (see diagram). There is a force resisting the motion of the block and 200 J of work is done against this force when the block has moved a distance of 5 m up the plane from rest. Find the speed of the block when it has moved a distance of 5 m up the plane from rest.
CAIE M1 2023 November Q4
4 A particle \(P\) of mass 0.2 kg lies at rest on a rough horizontal plane. A horizontal force of 1.2 N is applied to \(P\).
  1. Given that \(P\) is in limiting equilibrium, find the coefficient of friction between \(P\) and the plane.
  2. Given instead that the coefficient of friction between \(P\) and the plane is 0.3 , find the distance travelled by \(P\) in the third second of its motion.
CAIE M1 2023 November Q5
5 A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the speed of \(A\) when it reaches a height of 20 m above the ground.
    When \(A\) reaches a height of 20 m , it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed \(32.5 \mathrm {~ms} ^ { - 1 }\). In the collision between the two particles, \(B\) is brought to instantaneous rest.
  2. Show that the velocity of \(A\) immediately after the collision is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards.
  3. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground.
CAIE M1 2023 November Q6
6 A railway engine of mass 120000 kg is towing a coach of mass 60000 kg up a straight track inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.02\). There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000 N and there are constant resistances to motion of 22000 N on the engine and 13000 N on the coach.
  1. Find the acceleration of the engine and find the tension in the coupling.
    At an instant when the engine is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a section of track inclined upwards at an angle \(\beta\) to the horizontal. The power produced by the engine is now 4500000 W and, as a result, the engine maintains a constant speed.
  2. Assuming that the resistance forces remain unchanged, find the value of \(\beta\).
CAIE M1 2023 November Q7
7 A particle \(X\) travels in a straight line. The velocity of \(X\) at time \(t\) s after leaving a fixed point \(O\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = - 0.1 t ^ { 3 } + 1.8 t ^ { 2 } - 6 t + 5.6$$ The acceleration of \(X\) is zero at \(t = p\) and \(t = q\), where \(p < q\).
  1. Find the value of \(p\) and the value of \(q\).
    It is given that the velocity of \(X\) is zero at \(t = 14\).
  2. Find the velocities of \(X\) at \(t = p\) and at \(t = q\), and hence sketch the velocity-time graph for the motion of \(X\) for \(0 \leqslant t \leqslant 15\).
  3. Find the total distance travelled by \(X\) between \(t = 0\) and \(t = 15\).
    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
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
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
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
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
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
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
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 Q1
1 Two particles, of masses 1.8 kg and 1.2 kg , are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest. Find the magnitude of the acceleration of the particles and find the tension in the string.
\includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-03_2717_29_105_22}
CAIE M1 2024 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-03_293_638_258_717} A particle of mass 7.5 kg , starting from rest at \(A\), slides down an inclined plane \(A B\). The point \(B\) is 12.5 metres vertically below the level of \(A\), as shown in the diagram.
  1. Given that the plane is smooth, use an energy method to find the speed of the particle at \(B\).
  2. It is given instead that the plane is rough and the particle reaches \(B\) with a speed of \(8 \mathrm {~ms} ^ { - 1 }\). The plane is 25 m long and the constant frictional force has magnitude \(F \mathrm {~N}\). Find the value of \(F\).
    \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-04_725_655_251_641} Coplanar forces of magnitudes \(52 \mathrm {~N} , 39 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of \(P\) and \(\theta\).
    \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-04_2716_38_109_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-05_2716_29_107_22}
CAIE M1 2024 November Q4
4 A bus travels between two stops, \(A\) and \(B\). The bus starts from rest at \(A\) and accelerates at a constant rate of \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(16 \mathrm {~ms} ^ { - 1 }\). It then travels at this constant speed before decelerating at a constant rate of \(0.75 a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\). The total time for the journey is 240 s .
  1. Sketch the velocity-time graph for the bus's journey from \(A\) to \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-05_401_1198_479_434}
  2. Find an expression, in terms of \(a\), for the length of time that the bus is travelling with constant speed.
  3. Given that the distance from \(A\) to \(B\) is 3000 m , find the value of \(a\).
CAIE M1 2024 November Q5
5 A particle, \(A\), is projected vertically upwards from a point \(O\) with a speed of \(80 \mathrm {~ms} ^ { - 1 }\). One second later a second particle, \(B\), with the same mass as \(A\), is projected vertically upwards from \(O\) with a speed of \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(T\) s after the first particle is projected, the two particles collide and coalesce to form a particle \(C\).
  1. Show that \(T = 3.5\).
  2. Find the height above \(O\) at which the particles collide.
    \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-07_2723_33_99_22}
  3. Find the time from \(A\) being projected until \(C\) returns to \(O\).
    \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-08_415_912_246_580} A particle of mass 1.2 kg is placed on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 7 } { 25 }\). The particle is kept in equilibrium by a horizontal force of magnitude \(P \mathrm {~N}\) acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is 0.15 . Find the least possible value of \(P\).
    \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-08_2714_38_109_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-09_2726_35_97_20}
CAIE M1 2024 November Q7
3 marks
7 A car has mass 1200 kg . When the car is travelling at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), there is a resistive force of magnitude \(k v \mathrm {~N}\). The maximum power of the car's engine is 92.16 kW .
  1. The car travels along a straight level road.
    1. The car has a greatest possible constant speed of \(48 \mathrm {~ms} ^ { - 1 }\). Show that \(k = 40\).
    2. At an instant when its speed is \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the greatest possible acceleration of the car. [3]
      \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-10_2716_40_109_2009}
  2. The car now travels at a constant speed up a hill inclined at an angle of \(\sin ^ { - 1 } 0.15\) to the horizontal. Find the greatest possible speed of the car going up the hill.
CAIE M1 2024 November Q8
8 A particle \(P\) moves in a straight line, passing through a point \(O\) with velocity \(4.2 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after \(P\) passes \(O\), the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of \(P\) is given by \(a = 0.6 t - 2.7\). Find the distance \(P\) travels between the times at which it is at instantaneous rest.
\includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-12_2715_38_109_2009}
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-14_2716_37_108_2010}
CAIE M1 2024 November Q2
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
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
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
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.
CAIE M1 2024 November Q6
6 A particle, \(P\), travels in a straight line, starting from a point \(O\) with velocity \(6 \mathrm {~ms} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$\begin{array} { l l } a = - 1.5 t ^ { \frac { 1 } { 2 } } & \text { for } 0 \leqslant t \leqslant 1 ,
a = 1.5 t ^ { \frac { 1 } { 2 } } - 3 t ^ { - \frac { 1 } { 2 } } & \text { for } t > 1 . \end{array}$$
  1. Find the velocity of \(P\) at \(t = 1\).
  2. Given that there is no change in the velocity of \(P\) when \(t = 1\), find an expression for the velocity of \(P\) for \(t > 1\).
    \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-11_2725_35_99_20}
  3. Given that the velocity of \(P\) is positive for \(t \leqslant 4\), find the total distance travelled between \(t = 0\) and \(t = 4\).
    \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_723_762_248_653} Two particles, \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the bottom of a rough plane inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = 0.6\). Particle \(A\) lies on the plane, and particle \(B\) hangs vertically below the pulley, 0.25 m above horizontal ground. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between \(A\) and the plane is 1.125 . Particle \(A\) is released from rest.
  4. Find the tension in the string and the magnitude of the acceleration of the particles.
    \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_2716_38_109_2012}
  5. When \(B\) reaches the ground, it comes to rest. Find the total distance that \(A\) travels down the plane from when it is released until it comes to rest. You may assume that \(A\) does not reach the pulley.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-14_2715_31_106_2016}
CAIE M1 2024 November Q1
1 An athlete has mass \(m \mathrm {~kg}\) .The athlete runs along a horizontal road against a constant resistance force of magnitude 24 N .The total work done by the athlete in increasing his speed from \(5 \mathrm {~ms} ^ { - 1 }\) to \(6 \mathrm {~ms} ^ { - 1 }\) while running a distance of 50 metres is 1541 J . Find the value of \(m\) .
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-04_464_1116_247_478} Coplanar forces of magnitudes \(16 \mathrm {~N} , 12 \mathrm {~N} , 24 \mathrm {~N}\) and 8 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium.
CAIE M1 2024 November Q3
3 A car of mass 1600 kg travels up a slope inclined at an angle of \(\sin ^ { - 1 } 0.08\) to the horizontal. There is a constant resistance of magnitude 240 N acting on the car.
  1. It is given that the car travels at a constant speed of \(32 \mathrm {~ms} ^ { - 1 }\). Find the power of the engine of the car.
  2. Find the acceleration of the car when its speed is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at 95\% of the power found in (a).