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CAIE M1 2022 June Q3
5 marks Moderate -0.8
3 A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5 . A force of magnitude \(X \mathrm {~N}\), acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. \includegraphics[max width=\textwidth, alt={}, center]{213e26a8-3e4e-4dd4-b287-02e5925f3f47-06_849_807_255_669} Three coplanar forces of magnitudes \(20 \mathrm {~N} , 100 \mathrm {~N}\) and \(F \mathrm {~N}\) act at a point. The directions of these forces are shown in the diagram. Given that the three forces are in equilibrium, find \(F\) and \(\alpha\).
CAIE M1 2022 June Q5
9 marks Standard +0.3
5 Two racing cars \(A\) and \(B\) are at rest alongside each other at a point \(O\) on a straight horizontal test track. The mass of \(A\) is 1200 kg . The engine of \(A\) produces a constant driving force of 4500 N . When \(A\) arrives at a point \(P\) its speed is \(25 \mathrm {~ms} ^ { - 1 }\). The distance \(O P\) is \(d \mathrm {~m}\). The work done against the resistance force experienced by \(A\) between \(O\) and \(P\) is 75000 J .
  1. Show that \(d = 100\).
    Car \(B\) starts off at the same instant as car \(A\). The two cars arrive at \(P\) simultaneously and with the same speed. The engine of \(B\) produces a driving force of 3200 N and the car experiences a constant resistance to motion of 1200 N .
  2. Find the mass of \(B\).
  3. Find the steady speed which \(B\) can maintain when its engine is working at the same rate as it is at \(P\).
CAIE M1 2022 June Q6
10 marks Standard +0.3
6 A particle starts from a point \(O\) and moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is given by $$v = k \left( 3 t ^ { 2 } - 2 t ^ { 3 } \right)$$ where \(k\) is a constant.
  1. Verify that the particle returns to \(O\) when \(t = 2\).
  2. It is given that the acceleration of the particle is \(- 13.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the positive value of \(t\) at which \(v = 0\). Find \(k\) and hence find the total distance travelled in the first two seconds of motion.
CAIE M1 2022 June Q7
9 marks Challenging +1.2
7 Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal, and \(A\) is higher up the plane than \(B\). When the particles collide, the speeds of \(A\) and \(B\) are \(3 \mathrm {~ms} ^ { - 1 }\) and \(2 \mathrm {~ms} ^ { - 1 }\) respectively. In the collision between the particles, the speed of \(A\) is reduced to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the speed of \(B\) immediately after the collision.
    After the collision, when \(B\) has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. \(B\) hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by \(90 \%\). The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.
  2. Show that the speed of \(B\) immediately after it hits the barrier is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Hence find the speed of the combined particle immediately after the second collision between \(A\) and \(B\).
    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 2022 June Q1
5 marks Moderate -0.8
1 Small smooth spheres \(A\) and \(B\), of equal radii and of masses 5 kg and 3 kg respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(8.5 \mathrm {~ms} ^ { - 1 }\). The spheres collide and after the collision \(A\) continues to move in the same direction but with a quarter of the speed of \(B\).
  1. Find the speed of \(B\) after the collision.
  2. Find the loss of kinetic energy of the system due to the collision.
CAIE M1 2022 June Q2
6 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{4ee2568e-5902-442f-9af1-3663fa1d59c7-03_680_636_255_756} Coplanar forces of magnitudes \(60 \mathrm {~N} , 20 \mathrm {~N} , 16 \mathrm {~N}\) and 14 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the resultant force.
CAIE M1 2022 June Q3
7 marks Standard +0.3
3 Two particles \(A\) and \(B\), of masses 2.4 kg and 1.2 kg respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at a distance of 2.1 m above a horizontal plane and \(B\) is 1.5 m above the plane. The particles hang vertically and are released from rest. In the subsequent motion \(A\) reaches the plane and does not rebound and \(B\) does not reach the pulley.
  1. Show that the tension in the string before \(A\) reaches the plane is 16 N and find the magnitude of the acceleration of the particles before \(A\) reaches the plane.
  2. Find the greatest height of \(B\) above the plane.
CAIE M1 2022 June Q4
9 marks Moderate -0.3
4 A particle \(A\), moving along a straight horizontal track with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passes a fixed point \(O\). Four seconds later, another particle \(B\) passes \(O\), moving along a parallel track in the same direction as \(A\). Particle \(B\) has speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it passes \(O\) and has a constant deceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 } . B\) comes to rest when it returns to \(O\).
  1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) seconds after \(B\) passes \(O\).
  2. Find the values of \(t\) when the particles are the same distance from \(O\).
  3. On the given axes, sketch the displacement-time graphs for both particles, for values of \(t\) from 0 to 20 . \includegraphics[max width=\textwidth, alt={}, center]{4ee2568e-5902-442f-9af1-3663fa1d59c7-07_805_1259_1672_484} \includegraphics[max width=\textwidth, alt={}, center]{4ee2568e-5902-442f-9af1-3663fa1d59c7-08_467_583_255_781} A block of mass 12 kg is placed on a plane which is inclined at an angle of \(24 ^ { \circ }\) to the horizontal. A light string, making an angle of \(36 ^ { \circ }\) above a line of greatest slope, is attached to the block. The tension in the string is 65 N (see diagram). The coefficient of friction between the block and plane is \(\mu\). The block is in limiting equilibrium and is on the point of sliding up the plane. Find \(\mu\).
CAIE M1 2022 June Q6
8 marks Standard +0.3
6 A car of mass 900 kg is moving up a hill inclined at \(\sin ^ { - 1 } 0.12\) to the horizontal. The initial speed of the car is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After 12 s, the car has travelled 150 m up the hill and has speed \(16 \mathrm {~ms} ^ { - 1 }\). The engine of the car is working at a constant rate of 24 kW .
  1. Find the work done against the resistive forces during the 12 s .
    The car then travels along a straight horizontal road. There is a resistance to the motion of the car of \(( 1520 + 4 v ) \mathrm { N }\) 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 a constant rate of 32 kW .
  2. Find this speed.
CAIE M1 2022 June Q7
9 marks Standard +0.3
7 A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$\begin{array} { l l } v = 0.5 t & \text { for } 0 \leqslant t \leqslant 10 \\ v = 0.25 t ^ { 2 } - 8 t + 60 & \text { for } 10 \leqslant t \leqslant 20 \end{array}$$
  1. Show that there is an instantaneous change in the acceleration of the particle at \(t = 10\).
  2. Find the total distance covered by \(P\) in the interval \(0 \leqslant t \leqslant 20\).
    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 2022 June Q1
4 marks Moderate -0.8
1 Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(Q\) begins to move with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the speed of \(P\) after the collision.
    After the collision, \(Q\) moves directly towards a third particle \(R\), of mass \(m \mathrm {~kg}\), which is at rest on the plane. The two particles \(Q\) and \(R\) coalesce on impact and move with a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
  2. Find \(m\).
CAIE M1 2022 June Q2
5 marks Moderate -0.8
2 A particle \(P\) is projected vertically upwards from horizontal ground. \(P\) reaches a maximum height of 45 m . After reaching the ground, \(P\) comes to rest without rebounding.
  1. Find the speed at which \(P\) was projected.
  2. Find the total time for which the speed of \(P\) is at least \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2022 June Q3
5 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-04_824_1636_264_258} The displacement of a particle moving in a straight line is \(s\) metres at time \(t\) seconds after leaving a fixed point \(O\). The particle starts from rest and passes through points \(P , Q\) and \(R\), at times \(t = 5 , t = 10\) and \(t = 15\) respectively, and returns to \(O\) at time \(t = 20\). The distances \(O P , O Q\) and \(O R\) are 50 m , 150 m and 200 m respectively. The diagram shows a displacement-time graph which models the motion of the particle from \(t = 0\) to \(t = 20\). The graph consists of two curved segments \(A B\) and \(C D\) and two straight line segments \(B C\) and \(D E\).
  1. Find the speed of the particle between \(t = 5\) and \(t = 10\).
  2. Find the acceleration of the particle between \(t = 0\) and \(t = 5\), given that it is constant.
  3. Find the average speed of the particle during its motion. \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-06_483_880_258_630} The diagram shows a block of mass 10 kg suspended below a horizontal ceiling by two strings \(A C\) and \(B C\), of lengths 0.8 m and 0.6 m respectively, attached to fixed points on the ceiling. Angle \(A C B = 90 ^ { \circ }\). There is a horizontal force of magnitude \(F \mathrm {~N}\) acting on the block. The block is in equilibrium.
CAIE M1 2022 June Q5
8 marks Moderate -0.3
5 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and her bicycle is 70 kg . At an instant when the cyclist's speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude 30 N .
  1. Find the power developed by the cyclist.
    The cyclist comes to the top of a hill inclined at \(5 ^ { \circ }\) to the horizontal. The cyclist stops pedalling and freewheels down the hill (so that the cyclist is no longer supplying any power). The magnitude of the resistance force remains at 30 N . Over a distance of \(d \mathrm {~m}\), the speed of the cyclist increases from \(6 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
  2. Find the change in kinetic energy.
  3. Use an energy method to find \(d\). \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-10_725_785_260_680} 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 at \(B\) which is attached to two inclined planes. \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. \(Q\) lies on a plane \(B C\) which is inclined at \(30 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
CAIE M1 2022 June Q7
10 marks Standard +0.3
7 A particle \(P\) moves in a straight line through a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\), at time \(t \mathrm {~s}\) after passing \(O\), is given by $$v = \frac { 9 } { 4 } + \frac { b } { ( t + 1 ) ^ { 2 } } - c t ^ { 2 }$$ where \(b\) and \(c\) are positive constants. At \(t = 5\), the velocity of \(P\) is zero and its acceleration is \(- \frac { 13 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that \(b = 9\) and find the value of \(c\).
  2. Given that the velocity of \(P\) is zero only at \(t = 5\), find the distance travelled in the first 10 seconds of motion.
    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 June Q1
4 marks Standard +0.3
1 Two particles \(P\) and \(Q\), of masses \(m \mathrm {~kg}\) and 0.3 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(5 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(P\) moves with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the same direction as it was originally moving.
  1. Find, in terms of \(m\), the speed of \(Q\) after the collision.
    After this collision, \(Q\) moves directly towards a third particle \(R\), of mass 0.6 kg , which is at rest on the plane. \(Q\) is brought to rest in the collision with \(R\), and \(R\) begins to move with a speed of \(1.5 \mathrm {~ms} ^ { - 1 }\).
  2. Find the value of \(m\).
CAIE M1 2023 June Q2
6 marks Standard +0.3
2 A particle \(P\) of mass 0.4 kg is projected vertically upwards from horizontal ground with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the greatest height above the ground reached by \(P\).
    When \(P\) reaches the ground again, it bounces vertically upwards. At the first instant that it hits the ground, \(P\) loses 7.2 J of energy.
  2. Find the time between the first and second instants at which \(P\) hits the ground.
CAIE M1 2023 June Q3
4 marks Moderate -0.3
3 A particle moves in a straight line starting from rest. The displacement \(s m\) of the particle from a fixed point \(O\) on the line at time \(t \mathrm {~s}\) is given by $$s = t ^ { \frac { 5 } { 2 } } - \frac { 15 } { 4 } t ^ { \frac { 3 } { 2 } } + 6$$ Find the value of \(s\) when the particle is again at rest. \includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-06_730_1545_280_294} The velocity of a particle at time \(t \mathrm {~s}\) after leaving a fixed point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The diagram shows a velocity-time graph which models the motion of the particle. The graph consists of 5 straight line segments. The particle accelerates to a speed of \(0.9 \mathrm {~ms} ^ { - 1 }\) in a period of 3 s , then travels at constant speed for 6 s , and then comes instantaneously to rest 1 s later. The particle then moves back and returns to rest at \(O\) at time \(T \mathrm {~s}\).
  1. Find the distance travelled by the particle in the first 10 s of its motion.
  2. Given that \(T = 12\), find the minimum velocity of the particle.
  3. Given instead that the greatest speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(T\) and hence find the average speed of the particle for the whole of the motion. \includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-08_858_563_264_794} Four coplanar forces act at a point. The magnitudes of the forces are \(F \mathrm {~N} , 10 \mathrm {~N} , 50 \mathrm {~N}\) and 40 N . The directions of the forces are as shown in the diagram.
CAIE M1 2023 June Q7
11 marks Standard +0.3
7 A car of mass 1200 kg is travelling along a straight horizontal road. The power of the car's engine is constant and is equal to 16 kW . There is a constant resistance to motion of magnitude 500 N .
  1. Find the acceleration of the car at an instant when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Assuming that the power and the resistance forces remain unchanged, find the steady speed at which the car can travel.
    The car comes to the bottom of a straight hill of length 316 m , inclined at an angle to the horizontal of \(\sin ^ { - 1 } \left( \frac { 1 } { 60 } \right)\). The power remains constant at 16 kW , but the magnitude of the resistance force is no longer constant and changes such that the work done against the resistance force in ascending the hill is 128400 J . The time taken to ascend the hill is 15 s .
  3. Given that the car is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill, find its speed at the top of the hill.
    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 June Q1
3 marks Moderate -0.8
1 A particle of mass 1.6 kg is dropped from a height of 9 m above horizontal ground. The speed of the particle at the instant before hitting the ground is \(12 \mathrm {~ms} ^ { - 1 }\). Find the work done against air resistance.
CAIE M1 2023 June Q2
4 marks Easy -1.2
2 Two particles \(A\) and \(B\), of masses 3.2 kg and 2.4 kg respectively, lie on a smooth horizontal table. \(A\) moves towards \(B\) with a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and collides with \(B\), which is moving towards \(A\) with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the two particles come to rest.
  1. Find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-03_61_1569_495_328}
  2. Find the loss of kinetic energy of the system due to the collision.
CAIE M1 2023 June Q3
6 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-04_442_636_264_758} Coplanar forces of magnitudes \(30 \mathrm {~N} , 15 \mathrm {~N} , 33 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram, where \(\tan \alpha = \frac { 4 } { 3 }\). The system is in equilibrium.
  1. Show that \(\left( \frac { 14.4 } { 30 - P } \right) ^ { 2 } + \left( \frac { 28.8 } { P + 30 } \right) ^ { 2 } = 1\).
  2. Verify that \(P = 6\) satisfies this equation and find the value of \(\theta\).
CAIE M1 2023 June Q4
7 marks Moderate -0.3
4 An athlete of mass 84 kg is running along a straight road.
  1. Initially the road is horizontal and he runs at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The athlete produces a constant power of 60 W . Find the resistive force which acts on the athlete.
  2. The athlete then runs up a 150 m section of the road which is inclined at \(0.8 ^ { \circ }\) to the horizontal. The speed of the athlete at the start of this section of road is \(3 \mathrm {~ms} ^ { - 1 }\) and he now produces a constant driving force of 24 N . The total resistive force which acts on the athlete along this section of road has constant magnitude 13 N . Use an energy method to find the speed of the athlete at the end of the 150 m section of road.
CAIE M1 2023 June Q5
6 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-07_366_567_258_790} A particle of mass 0.6 kg is placed on a rough plane which is inclined at an angle of \(35 ^ { \circ }\) to the horizontal. 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 plane is 0.4 . Find the least possible value of \(P\).
CAIE M1 2023 June Q6
11 marks Standard +0.3
6 A particle \(P\) starts at rest and moves in a straight line from a point \(O\). At time \(t\) s after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = b t + c t ^ { \frac { 3 } { 2 } }\), where \(b\) and \(c\) are constants. \(P\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\) and has velocity \(13.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 9\).
  1. Show that \(b = 3\) and \(c = - 0.5\).
  2. Find the acceleration of \(P\) when \(t = 1\).
  3. Find the positive value of \(t\) when \(P\) is at instantaneous rest and find the distance of \(P\) from \(O\) at this instant.
  4. Find the speed of \(P\) at the instant it returns to \(O\).