Questions — CAIE M1 (786 questions)

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CAIE M1 2019 November Q6
11 marks Standard +0.3
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. [4]
  2. Show that the coefficient of friction between the block and the plane is 0.165, correct to 3 significant figures. [3]
  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. [4]
CAIE M1 2019 November Q7
11 marks Standard +0.3
\includegraphics{figure_7} 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.
  1. Find the tension in the string and the magnitude of the acceleration of the particles. [5]
\(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.
  1. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor. [2]
  2. 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. [4]
CAIE M1 Specimen Q1
4 marks Easy -1.2
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]
  2. Given that the time taken to raise the mass is 1.2 s, find the average power developed by the weightlifter. [2]
CAIE M1 Specimen Q2
6 marks Moderate -0.8
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° to the horizontal.
  1. Find the time taken for the particle to reach a speed of 2.5 m s\(^{-1}\). [3]
  2. Find the distance that the particle travels along the ground before it comes to rest. [3]
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.
CAIE M1 Specimen Q3
6 marks Standard +0.3
A lorry of mass 24 000 kg is travelling up a hill which is inclined at 3° 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 m s\(^{-1}\), its acceleration is 0.2 m s\(^{-2}\). Find the power developed by the lorry's engine. [4]
  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. [2]
CAIE M1 Specimen Q4
6 marks Standard +0.3
\includegraphics{figure_4} Blocks \(P\) and \(Q\), of mass \(m\) 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° 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. [6]
CAIE M1 Specimen Q5
8 marks Standard +0.3
\includegraphics{figure_5} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(AB\) of length 3 m. Three horizontal forces of magnitudes \(F\) N, 10 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 \(R\) N in the direction shown in the diagram.
  1. Find the values of \(F\) and \(R\). [5]
  2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of 11.7 m s\(^{-1}\). Find the mass of the bead. [3]
CAIE M1 Specimen Q6
10 marks Standard +0.3
A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in m s\(^{-1}\), at time \(t\) s after leaving \(O\) is given by $$v = 0.6t - 0.03t^2.$$
  1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time. [4]
  2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity. [6]
CAIE M1 Specimen Q7
10 marks Standard +0.3
A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration 0.5 m 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 \(AB\) is 210 m.
  1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). [5]
  2. Find the time that it takes from when the cyclist starts until the car overtakes her. [5]
24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration 4 m s\(^{-2}\) towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
CAIE M1 2014 June Q4
Standard +0.3
4 A particle \(P\) moves on a straight line, starting from rest at a point \(O\) of the line. The time after \(P\) starts to move is \(t \mathrm {~s}\), and the particle moves along the line with constant acceleration \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it passes through a point \(A\) at time \(t = 8\). After passing through \(A\) the velocity of \(P\) is \(\frac { 1 } { 2 } t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the acceleration of \(P\) immediately after it passes through \(A\). Hence show that the acceleration of \(P\) decreases by \(\frac { 1 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it passes through \(A\).
  2. Find the distance moved by \(P\) from \(t = 0\) to \(t = 27\).
CAIE M1 2014 June Q5
Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-3_343_691_254_725} A light inextensible rope has a block \(A\) of mass 5 kg attached at one end, and a block \(B\) of mass 16 kg attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Block \(A\) is held at rest at the bottom of the plane and block \(B\) hangs below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { \sqrt { 3 } }\). Block \(A\) is released from rest and the system starts to move. When each of the blocks has moved a distance of \(x \mathrm {~m}\) each has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down the gain in kinetic energy of the system in terms of \(v\).
  2. Find, in terms of \(x\),
    (a) the loss of gravitational potential energy of the system,
    (b) the work done against the frictional force.
  3. Show that \(21 v ^ { 2 } = 220 x\).