Questions — CAIE M1 (732 questions)

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CAIE M1 2024 June Q6
6 A particle moves in a straight line, starting from a point \(O\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v \mathrm {~ms} ^ { - 1 }\). It is given that \(\mathbf { v } = \mathrm { kt } ^ { \frac { 1 } { 2 } } - 2 \mathrm { t } - 8\), where \(k\) is a positive constant. The maximum velocity of the particle is \(4.5 \mathrm {~ms} ^ { - 1 }\).
  1. Show that \(k = 10\).
    1. Verify that \(v = 0\) when \(t = 1\) and \(t = 16\).
    2. Find the distance travelled by the particle in the first 16 s .
CAIE M1 2024 June Q7
7 A particle \(P\) of mass 0.2 kg is projected vertically upwards from horizontal ground with speed \(25 \mathrm {~ms} ^ { - 1 }\).
  1. Show that the speed of \(P\) when it reaches 20 m above the ground is \(15 \mathrm {~ms} ^ { - 1 }\).
    When \(P\) reaches 20 m above the ground it collides with a second particle \(Q\) of mass 0.1 kg which is moving downwards at \(20 \mathrm {~ms} ^ { - 1 } . P\) is brought to instantaneous rest in the collision.
  2. Find the velocity of \(Q\) immediately after the collision.
    When \(P\) reaches the ground it rebounds back directly upwards with half of the speed that it had immediately before hitting the ground.
  3. Find the height above the ground at which \(P\) and \(Q\) next collide.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE M1 2024 June Q1
1 A cyclist and bicycle have a total mass of 72 kg . The cyclist rides along a horizontal road against a total resistance force of 28 N . Find the total work done by the cyclist to increase his speed from \(8 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\) while travelling a distance of 100 metres.
CAIE M1 2024 June Q2
2 A particle \(P\) moves in a straight line. At time \(t\) s after leaving a point \(O\) on the line, \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\), where \(\mathrm { v } = 44 \mathrm { t } - 6 \mathrm { t } ^ { 2 } - 36\).
  1. Find the set of values of \(t\) for which the acceleration of the particle is positive.
  2. Find the two values of \(t\) at which \(P\) returns to \(O\).
    \includegraphics[max width=\textwidth, alt={}, center]{3eaf3652-ff91-4bae-9f20-83487d635612-04_714_796_248_635} Four coplanar forces of magnitude \(P \mathrm {~N} , 10 \mathrm {~N} , 16 \mathrm {~N}\) and 2 N act at a point in the directions shown in the diagram. It is given that the forces are in equilibrium. Find the values of \(\theta\) and \(P\).
CAIE M1 2024 June Q4
4 A car has mass 1400 kg . When the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\) the magnitude of the resistance to motion is \(\mathrm { kv } ^ { 2 } \mathrm {~N}\) where \(k\) is a constant.
  1. The car moves at a constant speed of \(24 \mathrm {~ms} ^ { - 1 }\) up a hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.12\). At this speed the magnitude of the resistance to motion is 480 N .
    1. Find the value of \(k\).
    2. Find the power of the car's engine.
  2. The car now moves at a constant speed on a straight level road. Given that its engine is working at 54 kW , find this speed.
    \includegraphics[max width=\textwidth, alt={}, center]{3eaf3652-ff91-4bae-9f20-83487d635612-06_542_923_251_571} A particle of mass 0.8 kg lies on a rough plane which is inclined at an angle of \(28 ^ { \circ }\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(T \mathrm {~N}\). This force acts at an angle of \(35 ^ { \circ }\) above a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.2 . Find the least and greatest possible values of \(T\).
CAIE M1 2024 June Q6
6 Three particles \(A , B\) and \(C\) of masses \(5 \mathrm {~kg} , 1 \mathrm {~kg}\) and 2 kg respectively lie at rest in that order on a straight smooth horizontal track \(X Y Z\). Initially \(A\) is at \(X , B\) is at \(Y\) and \(C\) is at \(Z\). Particle \(A\) is projected towards \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and at the same instant \(C\) is projected towards \(B\) with a speed of \(v \mathrm {~ms} ^ { - 1 }\). In the subsequent motion, \(A\) collides and coalesces with \(B\) to form particle \(D\). Particle \(D\) then collides and coalesces with \(C\) to form particle \(E\) and \(E\) moves towards \(Z\).
  1. Show that after the second collision the speed of \(E\) is \(\frac { 15 - v } { 4 } \mathrm {~ms} ^ { - 1 }\).
  2. The total loss of kinetic energy of the system due to the two collisions is 63 J . Use the result from (a) to show that \(v = 3\).
  3. It is given that the distance \(X Y\) is 36 m and the distance \(Y Z\) is 98 m .
    1. Find the time between the two collisions.
    2. Find the time between the instant that \(A\) is projected from \(X\) and the instant that \(E\) reaches \(Z\).
CAIE M1 2024 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{3eaf3652-ff91-4bae-9f20-83487d635612-10_621_908_248_580} Two particles \(P\) and \(Q\) of masses 2.5 kg and 0.5 kg respectively are connected by a light inextensible string that passes over a small smooth pulley fixed at the top of a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Particle \(P\) is on the plane and \(Q\) hangs below the pulley such that the level of \(Q\) is 2 m below the level of \(P\) (see diagram). Particle \(P\) is released from rest with the string taut and slides down the plane. The plane is rough with coefficient of friction 0.2 between the plane and \(P\).
  1. Find the acceleration of \(P\).
  2. Use an energy method to find the speed of the particles at the instant when they are at the same vertical height.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE M1 2020 March Q1
1 A lorry of mass 16000 kg is travelling along a straight horizontal road. The engine of the lorry is working at constant power. The work done by the driving force in 10 s is 750000 J .
  1. Find the power of the lorry's engine.
  2. There is a constant resistance force acting on the lorry of magnitude 2400 N . Find the acceleration of the lorry at an instant when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2020 March Q2
2 A particle \(P\) of mass 0.4 kg is on a rough horizontal floor. The coefficient of friction between \(P\) and the floor is \(\mu\). A force of magnitude 3 N is applied to \(P\) upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The particle is initially at rest and accelerates at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the time it takes for \(P\) to travel a distance of 1.44 m from its starting point.
  2. Find \(\mu\).
CAIE M1 2020 March Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-04_519_1018_260_561} The diagram shows the vertical cross-section of a surface. \(A , B\) and \(C\) are three points on the crosssection. The level of \(B\) is \(h \mathrm {~m}\) above the level of \(A\). The level of \(C\) is 0.5 m below the level of \(A\). A particle of mass 0.2 kg is projected up the slope from \(A\) with initial speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle remains in contact with the surface as it travels from \(A\) to \(C\).
  1. Given that the particle reaches \(B\) with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there is no resistance force, find \(h\).
  2. It is given instead that there is a resistance force and that the particle does 3.1 J of work against the resistance force as it travels from \(A\) to \(C\). Find the speed of the particle when it reaches \(C\).
CAIE M1 2020 March Q4
4 A cyclist travels along a straight road with constant acceleration. He passes through points \(A , B\) and \(C\). The cyclist takes 2 seconds to travel along each of the sections \(A B\) and \(B C\) and passes through \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is \(\frac { 4 } { 5 }\) of the distance \(B C\).
  1. Find the acceleration of the cyclist.
  2. Find \(A C\).
CAIE M1 2020 March Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-08_572_572_262_790} Coplanar forces, of magnitudes \(F \mathrm {~N} , 3 \mathrm {~N} , 6 \mathrm {~N}\) and 4 N , act at a point \(P\), as shown in the diagram.
  1. Given that \(\alpha = 60\), and that the resultant of the four forces is in the direction of the 3 N force, find \(F\).
  2. Given instead that the four forces are in equilibrium, find the values of \(F\) and \(\alpha\).
CAIE M1 2020 March Q6
6 On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass 1600 kg is towing a trailer of mass 700 kg along the track. The brakes are applied, resulting in a deceleration of \(12 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The braking force acts on the car only. In addition to the braking force there are constant resistance forces of 600 N on the car and of 200 N on the trailer.
  1. Find the magnitude of the force in the tow-bar.
  2. Find the braking force.
  3. At the instant when the brakes are applied, the car has speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At this instant the car is 17.5 m away from a stationary van, which is directly in front of the car. Show that the car hits the van at a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. After the collision, the van starts to move with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car and trailer continue moving in the same direction with speed \(2 \mathrm {~ms} ^ { - 1 }\). Find the mass of the van.
CAIE M1 2020 March Q7
7 A particle moves in a straight line through the point \(O\). The displacement of the particle from \(O\) at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$\begin{array} { l l } s = t ^ { 2 } - 3 t + 2 & \text { for } 0 \leqslant t \leqslant 6 ,
s = \frac { 24 } { t } - \frac { t ^ { 2 } } { 4 } + 25 & \text { for } t \geqslant 6 . \end{array}$$
  1. Find the value of \(t\) when the particle is instantaneously at rest during the first 6 seconds of its motion.
    At \(t = 6\), the particle hits a barrier at a point \(P\) and rebounds.
  2. Find the velocity with which the particle arrives at \(P\) and also the velocity with which the particle leaves \(P\).
  3. Find the total distance travelled by the particle in the first 10 seconds of its 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 2021 March Q1
1 Two particles \(P\) and \(Q\) of masses 0.2 kg and 0.3 kg respectively are free to move in a horizontal straight line on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant \(Q\) is projected towards \(P\) with speed \(1 \mathrm {~ms} ^ { - 1 } . Q\) comes to rest in the resulting collision. Find the speed of \(P\) after the collision.
CAIE M1 2021 March Q2
2 A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). There is a constant resistance force of magnitude 600 N . The power of the car's engine is 22500 W .
  1. Show that the speed of the car is \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The car, moving with speed \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), comes to a section of the hill which is inclined at \(2 ^ { \circ }\) to the horizontal.
  2. Given that the power and resistance force do not change, find the initial acceleration of the car up this section of the hill.
CAIE M1 2021 March Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-05_518_616_255_767} A particle \(Q\) of mass 0.2 kg is held in equilibrium by two light inextensible strings \(P Q\) and \(Q R . P\) is a fixed point on a vertical wall and \(R\) is a fixed point on a horizontal floor. The angles which strings \(P Q\) and \(Q R\) make with the horizontal are \(60 ^ { \circ }\) and \(30 ^ { \circ }\) respectively (see diagram). Find the tensions in the two strings.
CAIE M1 2021 March Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-06_661_1529_260_306} An elevator moves vertically, supported by a cable. The diagram shows a velocity-time graph which models the motion of the elevator. The graph consists of 7 straight line segments. The elevator accelerates upwards from rest to a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 1.5 s and then travels at this speed for 4.5 s , before decelerating to rest over a period of 1 s . The elevator then remains at rest for 6 s , before accelerating to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards over a period of 2 s . The elevator travels at this speed for a period of 5 s , before decelerating to rest over a period of 1.5 s .
  1. Find the acceleration of the elevator during the first 1.5 s .
  2. Given that the elevator starts and finishes its journey on the ground floor, find \(V\).
  3. The combined weight of the elevator and passengers on its upward journey is 1500 kg . Assuming that there is no resistance to motion, find the tension in the elevator cable on its upward journey when the elevator is decelerating.
CAIE M1 2021 March Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-08_286_661_260_742} A block of mass 5 kg is being pulled along a rough horizontal floor by a force of magnitude \(X \mathrm {~N}\) acting at \(30 ^ { \circ }\) above the horizontal (see diagram). The block starts from rest and travels 2 m in the first 5 s of its motion.
  1. Find the acceleration of the block.
  2. Given that the coefficient of friction between the block and the floor is 0.4 , find \(X\).
    The block is now placed on a part of the floor where the coefficient of friction between the block and the floor has a different value. The value of \(X\) is changed to 25, and the block is now in limiting equilibrium.
  3. Find the value of the coefficient of friction between the block and this part of the floor.
CAIE M1 2021 March Q6
6 A particle moves in a straight line. It starts from rest from a fixed point \(O\) on the line. Its velocity at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t ^ { 2 } - 8 t ^ { \frac { 3 } { 2 } } + 10 t\).
  1. Find the displacement of the particle from \(O\) when \(t = 1\).
  2. Show that the minimum velocity of the particle is \(- 125 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2021 March Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-12_439_1095_258_525} Two particles \(P\) and \(Q\) of masses 0.5 kg and \(m \mathrm {~kg}\) respectively are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the top of two inclined planes. The particles are initially at rest with \(P\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal and \(Q\) on a plane inclined at \(45 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 0.8 N is applied to \(P\) acting down the plane, causing \(P\) to move down the plane (see diagram).
  1. It is given that \(m = 0.3\), and that the plane on which \(Q\) rests is smooth. Find the tension in the string.
  2. It is given instead that the plane on which \(Q\) rests is rough, and that after each particle has moved a distance of 1 m , their speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction in this part of the motion is 0.5 J . Use an energy method to find the value of \(m\).
    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 March Q1
1 A crane is used to raise a block of mass 600 kg vertically upwards at a constant speed through a height of 15 m . There is a resistance to the motion of the block, which the crane does 10000 J of work to overcome.
  1. Find the total work done by the crane.
  2. Given that the average power exerted by the crane is 12.5 kW , find the total time for which the block is in motion.
CAIE M1 2022 March Q2
2 A particle \(P\) is projected vertically upwards from horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) reaches a maximum height of 20 m above the ground.
  1. Find the value of \(u\).
  2. Find the total time for which \(P\) is at least 15 m above the ground.
CAIE M1 2022 March Q3
3 A car of mass \(m \mathrm {~kg}\) is towing a trailer of mass 300 kg down a straight hill inclined at \(3 ^ { \circ }\) to the horizontal at a constant speed. There are resistance forces on the car and on the trailer, and the total work done against the resistance forces in a distance of 50 m is 40000 J . The engine of the car is doing no work and the tow-bar is light and rigid.
  1. Find the value of \(m\).
    The resistance force on the trailer is 200 N .
  2. Find the tension in the tow-bar between the car and the trailer.
CAIE M1 2022 March Q4
2 marks
4 The total mass of a cyclist and her bicycle is 70 kg . The cyclist is riding with constant power of 180 W up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). At an instant when the cyclist's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude \(F \mathrm {~N}\).
  1. Find the value of \(F\).
  2. Find the steady speed that the cyclist could maintain up the hill when working at this power. [2]