Questions — CAIE M1 (786 questions)

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CAIE M1 2016 June Q1
4 marks Moderate -0.3
1 A particle of mass 8 kg is pulled at a constant speed a distance of 20 m up a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal by a force acting along a line of greatest slope.
  1. Find the change in gravitational potential energy of the particle.
  2. The total work done against gravity and friction is 1146 J . Find the frictional force acting on the particle.
CAIE M1 2016 June Q2
5 marks Moderate -0.3
2 Alan starts walking from a point \(O\), at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along a horizontal path. Ben walks along the same path, also starting from \(O\). Ben starts from rest 5 s after Alan and accelerates at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . Ben then continues to walk at a constant speed until he is at the same point, \(P\), as Alan.
  1. Find how far Ben has travelled when he has been walking for 5 s and find his speed at this instant.
  2. Find the distance \(O P\).
CAIE M1 2016 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{aaf655c6-47f0-4f17-9a57-58aaf48728df-2_586_611_1171_767} The coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).
CAIE M1 2016 June Q4
7 marks Standard +0.3
4 A particle of mass 15 kg is stationary on a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.2 . A force of magnitude \(X \mathrm {~N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. Show that the least possible value of \(X\) is 23.1 , correct to 3 significant figures, and find the greatest possible value of \(X\).
CAIE M1 2016 June Q5
8 marks Moderate -0.3
5 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N .
  1. Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate of 20 kW .
  2. The car is travelling at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). Find the power of the car's engine.
  3. The car descends the same hill with the engine working at \(80 \%\) of the power found in part (ii). Find the acceleration of the car at an instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2016 June Q6
10 marks Standard +0.8
6 Two particles of masses 1.3 kg and 0.7 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held at the same vertical height with the string taut. The distance of each particle above a horizontal plane is 2 m , and the distance of each particle below the pulley is 4 m . The particles are released from rest.
  1. Find
    1. the tension in the string before the particle of mass 1.3 kg reaches the plane,
    2. the time taken for the particle of mass 1.3 kg to reach the plane.
    3. Find the greatest height of the particle of mass 0.7 kg above the plane.
CAIE M1 2016 June Q7
10 marks Standard +0.3
7 A particle \(P\) moves in a straight line. At time \(t \mathrm {~s}\), the displacement of \(P\) from \(O\) is \(s \mathrm {~m}\) and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 2\). When \(t = 1 , s = 7\) and when \(t = 3 , s = 29\).
  1. Find the set of values of \(t\) for which the particle is decelerating.
  2. Find \(s\) in terms of \(t\).
  3. Find the time when the velocity of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2017 June Q1
3 marks Moderate -0.8
1 A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done against air resistance.
CAIE M1 2017 June Q2
6 marks Moderate -0.3
2 A particle of mass 0.8 kg is projected with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a rough plane inclined at an angle of \(10 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.4 .
  1. Find the acceleration of the particle.
  2. Find the distance the particle moves up the plane before coming to rest.
CAIE M1 2017 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-04_442_584_255_778} Two light inextensible strings are attached to a particle of weight 25 N . The strings pass over two smooth fixed pulleys and have particles of weights \(A \mathrm {~N}\) and \(B \mathrm {~N}\) hanging vertically at their ends. The sloping parts of the strings make angles of \(30 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the vertical (see diagram). The system is in equilibrium. Find the values of \(A\) and \(B\).
CAIE M1 2017 June Q4
6 marks Moderate -0.3
4 A car of mass 800 kg is moving up a hill inclined at \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta = 0.15\). The initial speed of the car is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Twelve seconds later the car has travelled 120 m up the hill and has speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the change in the kinetic energy and the change in gravitational potential energy of the car.
  2. The engine of the car is working at a constant rate of 32 kW . Find the total work done against the resistive forces during the twelve seconds.
CAIE M1 2017 June Q5
7 marks Moderate -0.3
5 A particle \(P\) moves in a straight line \(A B C D\) with constant deceleration. The velocities of \(P\) at \(A , B\) and \(C\) are \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  1. Find the ratio of distances \(A B : B C\).
  2. The particle comes to rest at \(D\). Given that the distance \(A D\) is 80 m , find the distance \(B C\).
CAIE M1 2017 June Q6
10 marks Standard +0.3
6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = q t + r t ^ { 2 }\), where \(q\) and \(r\) are constants. The particle has velocity \(4 \mathrm {~ms} ^ { - 1 }\) when \(t = 1\) and when \(t = 2\).
  1. Show that, when \(t = 0.5\), the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    ............................................................................................................................... .
  2. Find the values of \(t\) when \(P\) is at instantaneous rest.
  3. The particle is at \(O\) when \(t = 3\). Find the distance of \(P\) from \(O\) when \(t = 0\).
CAIE M1 2017 June Q7
12 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-10_374_762_259_688} As shown in the diagram, a particle \(A\) of mass 0.8 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal and a particle \(B\) of mass 1.2 kg lies on a plane inclined at an angle of \(60 ^ { \circ }\) 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 parallel to lines of greatest slope of the respective planes. The particles are released from rest with both parts of the string taut.
  1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
  2. It is given instead that both planes are rough, with the same coefficient of friction, \(\mu\), for both particles. Find the value of \(\mu\) for which the system is in limiting equilibrium.
CAIE M1 2017 June Q1
5 marks Moderate -0.8
1 A man pushes a wheelbarrow of mass 25 kg along a horizontal road with a constant force of magnitude 35 N at an angle of \(20 ^ { \circ }\) below the horizontal. There is a constant resistance to motion of 15 N . The wheelbarrow moves a distance of 12 m from rest.
  1. Find the work done by the man.
  2. Find the speed attained by the wheelbarrow after 12 m .
CAIE M1 2017 June Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-03_522_604_262_769} The four coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).
CAIE M1 2017 June Q3
6 marks Standard +0.3
3 A train travels between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and accelerates at a constant rate for \(T\) s until it reaches a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at this constant speed before decelerating at a constant rate, coming to rest at \(B\). The magnitude of the train's deceleration is twice the magnitude of its acceleration. The total time taken for the journey is 180 s .
  1. Sketch the velocity-time graph for the train's journey from \(A\) to \(B\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-04_496_857_516_685}
  2. Find an expression, in terms of \(T\), for the length of time for which the train is travelling with constant speed.
  3. The distance from \(A\) to \(B\) is 3300 m . Find how far the train travels while it is decelerating.
CAIE M1 2017 June Q4
6 marks Moderate -0.3
4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by \(v = ( 2 t - 5 ) ^ { 3 }\).
  1. Find the values of \(t\) when the acceleration of \(P\) is \(54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find an expression for the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\).
CAIE M1 2017 June Q5
6 marks Standard +0.3
5 A particle is projected vertically upwards from a point \(O\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds later a second particle is projected vertically upwards from \(O\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after the second particle is projected, the two particles collide.
  1. Find \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-06_65_1569_488_328}
  2. Hence find the height above \(O\) at which the particles collide.
CAIE M1 2017 June Q6
8 marks Moderate -0.3
6 A car of mass 1200 kg is travelling along a horizontal road.
  1. It is given that there is a constant resistance to motion.
    1. The engine of the car is working at 16 kW while the car is travelling at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the resistance to motion.
    2. The power is now increased to 22.5 kW . Find the acceleration of the car at the instant it is travelling at a speed of \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    3. It is given instead that the resistance to motion of the car is \(( 590 + 2 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels at a constant speed with the engine working at 16 kW . Find this speed.
CAIE M1 2017 June Q7
14 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-10_336_803_258_671} Two particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and 4 kg respectively are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(A\) is on a rough fixed slope which is at an angle of \(30 ^ { \circ }\) to the horizontal ground. Particle \(B\) hangs vertically below the pulley and is 0.5 m above the ground (see diagram). The coefficient of friction between the slope and particle \(A\) is 0.2 .
  1. In the case where the system is in equilibrium with particle \(A\) on the point of moving directly up the slope, show that \(m = 5.94\), correct to 3 significant figures.
  2. In the case where \(m = 3\), 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.
CAIE M1 2018 June Q1
4 marks Moderate -0.3
1 A man has mass 80 kg . He runs along a horizontal road against a constant resistance force of magnitude \(P \mathrm {~N}\). The total work done by the man in increasing his speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while running a distance of 60 metres is 1200 J . Find the value of \(P\).
CAIE M1 2018 June Q2
4 marks Standard +0.3
2 A train of mass 240000 kg travels up a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance of magnitude 18000 N acting on the train. At an instant when the speed of the train is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its deceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power of the engine of the train.
CAIE M1 2018 June Q3
4 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{16640429-198d-4ea9-a2f6-6e2ef6ac1b4a-05_535_616_260_762} The three coplanar forces shown in the diagram have magnitudes \(3 \mathrm {~N} , 2 \mathrm {~N}\) and \(P \mathrm {~N}\). Given that the three forces are in equilibrium, find the values of \(\theta\) and \(P\).
CAIE M1 2018 June Q4
9 marks Moderate -0.3
4 A particle \(P\) moves in a straight line \(A B C D\) with constant acceleration. The distances \(A B\) and \(B C\) are 100 m and 148 m respectively. The particle takes 4 s to travel from \(A\) to \(B\) and also takes 4 s to travel from \(B\) to \(C\).
  1. Show that the acceleration of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the speed of \(P\) at \(A\).
  2. \(P\) reaches \(D\) with a speed of \(61 \mathrm {~ms} ^ { - 1 }\). Find the distance \(C D\).