Questions M1 (1912 questions)

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CAIE M1 2022 June Q5
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
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
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
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
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
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
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
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
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
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
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
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 \(O\).
CAIE M1 2023 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-10_551_776_260_689} Two particles \(P\) and \(Q\), of masses 2 kg and 0.25 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(P\) is on an inclined plane at an angle of \(30 ^ { \circ }\) to the horizontal. Particle \(Q\) hangs below the pulley. Three points \(A , B\) and \(C\) lie on a line of greatest slope of the plane with \(A B = 0.8 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram). Particle \(P\) is released from rest at \(A\) with the string taut and slides down the plane. During the motion of \(P\) from \(A\) to \(C , Q\) does not reach the pulley. The part of the plane from \(A\) to \(B\) is rough, with coefficient of friction 0.3 between the plane and \(P\). The part of the plane from \(B\) to \(C\) is smooth.
    1. Find the acceleration of \(P\) between \(A\) and \(B\).
    2. Hence, find the speed of \(P\) at \(C\).
  2. Find the time taken for \(P\) to travel from \(A\) to \(C\).
    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
1 Two particles \(P\) and \(Q\), of masses 0.1 kg and 0.4 kg respectively, are free to move on a smooth horizontal plane. Particle \(P\) is projected with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After \(P\) and \(Q\) collide, the speeds of \(P\) and \(Q\) are equal. Find the two possible values of the speed of \(P\) after the collision.
CAIE M1 2023 June Q2
2 A car of mass 1500 kg is towing a trailer of mass \(m \mathrm {~kg}\) along a straight horizontal road. The car and the trailer are connected by a tow-bar which is horizontal, light and rigid. There is a resistance force of \(F \mathrm {~N}\) on the car and a resistance force of 200 N on the trailer. The driving force of the car's engine is 3200 N , the acceleration of the car is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and the tension in the tow-bar is 300 N . Find the value of \(m\) and the value of \(F\).
CAIE M1 2023 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-04_337_661_262_740} A smooth ring \(R\) of mass 0.2 kg is threaded on a light string \(A R B\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(A B R = 90 ^ { \circ }\). The angle between the part \(A R\) of the string and the vertical is \(60 ^ { \circ }\). The ring is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), acting on the ring in a direction perpendicular to \(A R\) (see diagram). Calculate the tension in the string and the value of \(X\).
CAIE M1 2023 June Q4
4 A lorry of mass 15000 kg moves on a straight horizontal road in the direction from \(A\) to \(B\). It passes \(A\) and \(B\) with speeds \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The power of the lorry's engine is constant and there is a constant resistance to motion of magnitude 6000 N . The acceleration of the lorry at \(B\) is 0.5 times the acceleration of the lorry at \(A\).
  1. Show that the power of the lorry's engine is 200 kW , and hence find the acceleration of the lorry when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The lorry begins to ascend a straight hill inclined at \(1 ^ { \circ }\) to the horizontal. It is given that the power of the lorry's engine and the resistance force do not change.
  2. Find the steady speed up the hill that the lorry could maintain.
CAIE M1 2023 June Q5
5 A particle starts from rest from a point \(O\) and moves in a straight line. The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = k t ^ { \frac { 1 } { 2 } }\) for \(0 \leqslant t \leqslant 9\) and where \(k\) is a constant. The velocity of the particle at \(t = 9\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 0.1\).
    For \(t > 9\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle is given by \(v = 0.2 ( t - 9 ) ^ { 2 } + 1.8\).
  2. Show that the distance travelled in the first 9 seconds is one tenth of the distance travelled between \(t = 9\) and \(t = 18\).
  3. Find the greatest acceleration of the particle during the first 10 seconds of its motion.
CAIE M1 2023 June Q6
6 An elevator is pulled vertically upwards by a cable. The elevator accelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s , then travels at constant speed for 25 s . The elevator then decelerates at \(0.2 \mathrm {~ms} ^ { - 2 }\) until it comes to rest.
  1. Find the greatest speed of the elevator and hence draw a velocity-time graph for the motion of the elevator.
  2. Find the total distance travelled by the elevator.
    The mass of the elevator is 1200 kg and there is a crate of mass \(m \mathrm {~kg}\) resting on the floor of the elevator.
  3. Given that the tension in the cable when the elevator is decelerating is 12250 N , find the value of \(m\).
  4. Find the greatest magnitude of the force exerted on the crate by the floor of the elevator, and state its direction.
CAIE M1 2023 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-10_525_885_264_625} The diagram shows the vertical cross-section \(X Y Z\) of a rough slide. The section \(Y Z\) is a straight line of length 2 m inclined at an angle of \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The section \(Y Z\) is tangential to the curved section \(X Y\) at \(Y\), and \(X\) is 1.8 m above the level of \(Y\). A child of mass 25 kg slides down the slide, starting from rest at \(X\). The work done by the child against the resistance force in moving from \(X\) to \(Y\) is 50 J .
  1. Find the speed of the child at \(Y\).
    It is given that the child comes to rest at \(Z\).
  2. Use an energy method to find the coefficient of friction between the child and \(Y Z\), giving your answer as a fraction in its simplest form.
    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 2024 June Q1
1 A car starts from rest and accelerates at \(2 \mathrm {~ms} ^ { - 2 }\) for 10 s . It then travels at a constant speed for 30 s . The car then uniformly decelerates to rest over a period of 20 s .
  1. Sketch a velocity-time graph for the motion of the car.
    \includegraphics[max width=\textwidth, alt={}, center]{2af7fd9a-aa78-4d77-aa4e-c01604c8b0ae-03_762_1081_447_493}
  2. Find the total distance travelled by the car.
CAIE M1 2024 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{2af7fd9a-aa78-4d77-aa4e-c01604c8b0ae-04_558_606_276_715} Two forces of magnitudes 20 N and \(F \mathrm {~N}\) act at a point \(P\) in the directions shown in the diagram.
  1. Given that the resultant force has no component in the \(y\)-direction, calculate the value of \(F\).
  2. Given instead that \(F = 10\), find the magnitude and direction of the resultant force.
CAIE M1 2024 June Q3
3 A train of mass 180000 kg ascends a straight hill of length 1.5 km , inclined at an angle of \(1.5 ^ { \circ }\) to the horizontal. As it ascends the hill, the total work done to overcome the resistance to motion is 12000 kJ and the speed of the train decreases from \(45 \mathrm {~ms} ^ { - 1 }\) to \(40 \mathrm {~ms} ^ { - 1 }\). Find the work done by the engine of the train as it ascends the hill, giving your answer in kJ .
CAIE M1 2024 June Q4
4 A car of mass 1700 kg is pulling a trailer of mass 300 kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of 400 N on the car and 150 N on the trailer. The power of the car's engine is 14000 W . Find the acceleration of the car and the tension in the cable when the speed is \(20 \mathrm {~ms} ^ { - 1 }\).
CAIE M1 2024 June Q5
5 A straight slope of length 60 m is inclined at an angle of \(12 ^ { \circ }\) to the horizontal. A bobsled starts at the top of the slope with a speed of \(5 \mathrm {~ms} ^ { - 1 }\). The bobsled slides directly down the slope.
  1. It is given that there is no resistance to the bobsled's motion. Find its speed when it reaches the bottom of the slope.
  2. It is given instead that the coefficient of friction between the bobsled and the slope is 0.03 . Find the time that it takes for the bobsled to reach the bottom of the slope.