Questions — CAIE M1 (732 questions)

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CAIE M1 2019 November Q2
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
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
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
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
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
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
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
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
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
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
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 M1 2024 June Q1
1 Two particles \(P\) and \(Q\) of masses 0.2 kg and 0.5 kg respectively are at rest on a smooth horizontal plane. Particle \(P\) is projected with a speed \(6 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(P\) moves with a speed of \(1 \mathrm {~ms} ^ { - 1 }\). Find the two possible speeds of \(Q\) after the collision.
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-02_2716_35_143_2012}
CAIE M1 2024 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-03_721_622_296_724} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a vertical wall. The particle is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), perpendicular to the string, with the string taut and making an angle of \(30 ^ { \circ }\) with the wall (see diagram). Find the tension in the string and the value of \(X\).
CAIE M1 2024 June Q3
3 A car travels along a straight road with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\), where \(a > 0\). The car passes through points \(A , B\) and \(C\) in that order. The speed of the car at \(A\) is \(u \mathrm {~ms} ^ { - 1 }\) in the direction \(A B\). The distance \(B C\) is twice the distance \(A B\). The car takes 8 seconds to travel from \(A\) to \(B\) and 10 seconds to travel from \(B\) to \(C\).
  1. Find \(u\) in terms of \(a\).
  2. Find the speed of the car at \(C\) in terms of \(a\).
CAIE M1 2024 June Q4
4 A particle travels in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving a point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = k t ^ { 2 } - 4 t + 3$$ The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that \(v > 0\) in the first 2s of its motion.
  1. Find the value of \(k\).
  2. Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.
CAIE M1 2024 June Q5
5 A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of \(\theta\) to the horizontal where \(\sin \theta = 0.05\). The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.
  1. It is given that the tension in the tow-bar is 450 N . Find the acceleration of the trailer and the driving force of the van's engine.
    On another occasion, the van and trailer ascend a straight hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.09\). The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances to motion are unchanged.
    1. Find the acceleration of the van and the tension in the tow-bar.
    2. Find the speed of the van when it has travelled a distance of 375 m up the hill.
CAIE M1 2024 June Q6
6 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg . There is a constant resistance force of magnitude 32 N to the cyclist's motion. At an instant when she is travelling at \(7 \mathrm {~ms} ^ { - 1 }\), her acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).
  1. Find the power output of the cyclist.
  2. Find the steady speed that the cyclist can maintain if her power output and the resistance force are both unchanged.
    \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-08_2718_35_141_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-09_2724_35_136_20} The cyclist later descends a straight hill of length 32.2 m , inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 20 } \right)\) to the horizontal. Her power output is now 120 W , and the resistance force now has variable magnitude such that the work done against this force in descending the hill is 1128 J . The time taken to descend the hill is 4 s .
  3. Given that the speed of the cyclist at the top of the hill is \(7.5 \mathrm {~ms} ^ { - 1 }\), find her speed at the bottom of the hill.
CAIE M1 2024 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_323_1308_292_376} The diagram shows a track \(A B C D\) which lies in a vertical plane. The section \(A B\) is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is smooth. The section \(B C\) is a horizontal straight line and is rough. The section CD is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is rough. The lengths \(A B , B C\) and \(C D\) are each 2 m . A particle is released from rest at \(A\). The coefficient of friction between the particle and both \(B C\) and \(C D\) is \(\mu\). There is no change in the speed of the particle when it passes through either of the points \(B\) or \(C\).
  1. It is given that \(\mu = 0.1\). Find the distance which the particle has moved up the section \(C D\) when its speed is \(1 \mathrm {~ms} ^ { - 1 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_2716_33_143_2014}
  2. It is given instead that with a different value of \(\mu\) the particle travels 1 m up the track from \(C\) before it comes instantaneously to rest. Find the value of \(\mu\) and the speed of the particle at the instant that it passes \(C\) for the second time.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE M1 2022 June Q4
  1. In the case where \(F = 20\), find the tensions in each of the strings.
  2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.
CAIE M1 2022 June Q6
  1. It is given that the plane \(B C\) is smooth and that the particles are released from rest. Find the tension in the string and the magnitude of the acceleration of the particles.
  2. It is given instead that the plane \(B C\) is rough. A force of magnitude 3 N is applied to \(Q\) directly up the plane along a line of greatest slope of the plane. Find the least value of the coefficient of friction between \(Q\) and the plane \(B C\) for which the particles remain at rest.
CAIE M1 2023 June Q4
  1. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\).
  2. Given instead that \(F = 10 \sqrt { 2 }\) and \(\theta = 45\), find the direction and the exact magnitude the resultant force.
    \includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-10_518_627_264_756} Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. Particle \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. Particle \(Q\) lies on a plane \(B C\) which is inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
  3. It is given that \(\theta = 60\), the plane \(B C\) is rough and the coefficient of friction between \(Q\) and the plane \(B C\) is 0.7 . The particles are released from rest. Determine whether the particles move.
  4. It is given instead that the plane \(B C\) is smooth. The particles are released from rest and in the subsequent motion the tension in the string is \(( \sqrt { 3 } - 1 ) \mathrm { N }\). Find the magnitude of the acceleration of \(P\) as it moves on the plane, and find the value of \(\theta\).
CAIE M1 2023 March Q5
  1. Find the magnitude of the force in each of the struts \(A D\) and \(B D\).
    A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the block in a direction parallel to \(A B\).
  2. Find the value of \(F\) for which the magnitude of the force in the strut \(A D\) is zero.
    \includegraphics[max width=\textwidth, alt={}, center]{b2cd1b68-523f-40c3-8a51-acb2b55ae8c0-08_456_782_260_687} A block \(B\), of mass 2 kg , lies on a rough inclined plane sloping at \(30 ^ { \circ }\) to the horizontal. A light rope, inclined at an angle of \(20 ^ { \circ }\) above a line of greatest slope, is attached to \(B\). The tension in the rope is \(T \mathrm {~N}\). There is a friction force of \(F \mathrm {~N}\) acting on \(B\) (see diagram). The coefficient of friction between \(B\) and the plane is \(\mu\).
  3. It is given that \(F = 5\) and that the acceleration of \(B\) up the plane is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Find the value of \(T\).
    2. Find the value of \(\mu\).
  4. It is given instead that \(\mu = 0.8\) and \(T = 15\). Determine whether \(B\) will move up the plane.
CAIE M1 2011 June Q4
  1. Make a rough copy of the diagram and shade the region whose area represents the displacement of \(P\) from \(X\) at the instant when \(Q\) starts. It is given that \(P\) has travelled 70 m at the instant when \(Q\) starts.
  2. Find the value of \(T\).
  3. Find the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. Sketch a single diagram showing the displacement-time graphs for both \(P\) and \(Q\), with values shown on the \(t\)-axis at which the speed of either particle changes.
CAIE M1 2015 June Q6
  1. Find the value of \(h\).
  2. Find the value of \(m\), and find also the tension in the string while \(Q\) is moving.
  3. The string is slack while \(Q\) is at rest on the ground. Find the total time from the instant that \(P\) is released until the string becomes taut again.
CAIE M1 2019 June Q4
  1. Show that, before the string breaks, the magnitude of the acceleration of each particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. Find the difference in the times that it takes the particles to hit the ground.