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CAIE M1 2021 November Q6
11 marks Standard +0.3
6 A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest 14 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by $$\begin{array} { l l } v = p t ^ { 2 } - q t & 0 \leqslant t \leqslant 6 \\ v = 63 - 4.5 t & 6 \leqslant t \leqslant 14 \end{array}$$ where \(p\) and \(q\) are positive constants.
The acceleration of \(P\) is zero when \(t = 2\).
  1. Given that there are no instantaneous changes in velocity, find \(p\) and \(q\).
  2. Sketch the velocity-time graph.
  3. Find the total distance travelled by \(P\) during the 14 s . \includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-10_326_1109_255_520} Two particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are connected by a light inextensible string. Particle \(B\) is on a smooth fixed plane which is at an angle of \(18 ^ { \circ }\) to horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle \(A\) hangs vertically below the pulley and is 0.45 m above the ground (see diagram). The system is released from rest with the string taut. When \(A\) reaches the ground, the string breaks. Find the total distance travelled by \(B\) before coming to instantaneous rest. You may assume that \(B\) does not reach the pulley.
    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 November Q1
4 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-02_611_1351_260_397} The diagram shows a velocity-time graph which models the motion of a car. The graph consists of six straight line segments. The car accelerates from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 5 s , and then travels at this speed for a further 20 s . The car then decelerates to a speed of \(6 \mathrm {~ms} ^ { - 1 }\) over a period of 5 s . This speed is maintained for a further \(( T - 30 ) \mathrm { s }\). The car then accelerates again to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(( 50 - T ) \mathrm { s }\), before decelerating to rest over a period of 10 s .
  1. Given that during the two stages of the motion when the car is accelerating, the accelerations are equal, find the value of \(T\).
  2. Find the total distance travelled by the car during the motion.
CAIE M1 2021 November Q2
6 marks Standard +0.3
2 A van of mass 3600 kg is towing a trailer of mass 1200 kg along a straight horizontal road using a light horizontal rope. There are resistance forces of 700 N on the van and 300 N on the trailer.
  1. The driving force exerted by the van is 2500 N . Find the tension in the rope.
    The driving force is now removed and the van driver applies a braking force which acts only on the van. The resistance forces remain unchanged.
  2. Find the least possible value of the braking force which will cause the rope to become slack.
CAIE M1 2021 November Q3
5 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-04_416_792_260_674} The diagram shows a semi-circular track \(A B C\) of radius 1.8 m which is fixed in a vertical plane. The points \(A\) and \(C\) are at the same horizontal level and the point \(B\) is at the bottom of the track. The section \(A B\) is smooth and the section \(B C\) is rough. A small block is released from rest at \(A\).
  1. Show that the speed of the block at \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The block comes to instantaneous rest for the first time at a height of 1.2 m above the level of \(B\). The work done against the resistance force during the motion of the block from \(B\) to this point is 4.5 J .
  2. Find the mass of the block.
CAIE M1 2021 November Q4
7 marks Moderate -0.3
4 A cyclist starts from rest at a point \(A\) and travels along a straight road \(A B\), coming to rest at \(B\). The displacement of the cyclist from \(A\) at time \(t \mathrm {~s}\) after the start is \(s \mathrm {~m}\), where $$s = 0.004 \left( 75 t ^ { 2 } - t ^ { 3 } \right)$$
  1. Show that the distance \(A B\) is 250 m .
  2. Find the maximum velocity of the cyclist.
CAIE M1 2021 November Q5
7 marks Standard +0.3
5 A railway engine of mass 75000 kg is moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.01\). The engine is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine is working at 960 kW . There is a constant force resisting the motion of the engine.
  1. Find the resistance force.
    The engine comes to a section of track which is horizontal. At the start of the section the engine is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the power of the engine is now reduced to 900 kW . The resistance to motion is no longer constant, but in the next 60 s the work done against the resistance force is 46500 kJ .
  2. Find the speed of the engine at the end of the 60 s .
CAIE M1 2021 November Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-08_412_588_260_776} A block of mass 5 kg is held in equilibrium near a vertical wall by two light strings and a horizontal force of magnitude \(X \mathrm {~N}\), as shown in the diagram. The two strings are both inclined at \(60 ^ { \circ }\) to the vertical.
  1. Given that \(X = 100\), find the tension in the lower string.
  2. Find the least value of \(X\) for which the block remains in equilibrium in the position shown. [4]
CAIE M1 2021 November Q7
13 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-10_501_416_262_861} Particles \(P\) and \(Q\) have masses \(m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively. The particles are initially held at rest 6.4 m apart on the same line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.8\) (see diagram). Particle \(P\) is released from rest and slides down the line of greatest slope. Simultaneously, particle \(Q\) is projected up the same line of greatest slope at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between each particle and the plane is 0.6 .
  1. Show that the acceleration of \(Q\) up the plane is \(- 11.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the time for which the particles are in motion before they collide.
  3. The particles coalesce on impact. Find the speed of the combined particle immediately after the impact.
    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 November Q1
5 marks Moderate -0.5
1 \includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-03_471_613_254_766} A metal post is driven vertically into the ground by dropping a heavy object onto it from above. The mass of the object is 120 kg and the mass of the post is 40 kg (see diagram). The object hits the post with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and remains in contact with it after the impact.
  1. Calculate the speed with which the combined post and object moves immediately after the impact.
  2. There is a constant force resisting the motion of magnitude 4800 N . Calculate the distance the post is driven into the ground.
CAIE M1 2021 November Q2
7 marks Moderate -0.3
2 A particle of mass 8 kg is suspended in equilibrium by two light inextensible strings which make angles of \(60 ^ { \circ }\) and \(45 ^ { \circ }\) above the horizontal.
  1. Draw a diagram showing the forces acting on the particle.
  2. Find the tensions in the strings.
CAIE M1 2021 November Q3
6 marks Moderate -0.8
3 A ball of mass 1.6 kg is released from rest at a point 5 m above horizontal ground. When the ball hits the ground it instantaneously loses 8 J of kinetic energy and starts to move upwards.
  1. Use an energy method to find the greatest height that the ball reaches after hitting the ground.
  2. Find the total time taken, from the initial release of the ball until it reaches this greatest height.
CAIE M1 2021 November Q4
9 marks Moderate -0.3
4 A car of mass 1400 kg is moving on a straight road against a constant force of 1250 N resisting the motion.
  1. The car moves along a horizontal section of the road at a constant speed of \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Calculate the work done against the resisting force during the first 8 seconds.
    2. Calculate, in kW , the power developed by the engine of the car.
    3. Given that this power is suddenly increased by 12 kW , find the instantaneous acceleration of the car.
  2. The car now travels at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a section of the road inclined at \(\theta ^ { \circ }\) to the horizontal, with the engine working at 64 kW . Find the value of \(\theta\).
CAIE M1 2021 November Q5
11 marks Standard +0.3
5 A particle \(P\) moves in a straight line, starting from rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\) the acceleration of \(P\) is \(k \left( 16 - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(k\) is a positive constant, and the displacement from \(O\) is \(s \mathrm {~m}\). The velocity of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\).
  1. Show that \(s = \frac { 1 } { 64 } t ^ { 2 } \left( 96 - t ^ { 2 } \right)\).
  2. Find the speed of \(P\) at the instant that it returns to \(O\).
  3. Find the maximum displacement of the particle from \(O\).
CAIE M1 2021 November Q6
12 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-10_451_1315_258_415} The diagram shows a particle of mass 5 kg on a rough horizontal table, and two light inextensible strings attached to it passing over smooth pulleys fixed at the edges of the table. Particles of masses 4 kg and 6 kg hang freely at the ends of the strings. The particle of mass 6 kg is 0.5 m above the ground. The system is in limiting equilibrium.
  1. Show that the coefficient of friction between the 5 kg particle and the table is 0.4 .
    The 6 kg particle is now replaced by a particle of mass 8 kg and the system is released from rest.
  2. Find the acceleration of the 4 kg particle and the tensions in the strings.
  3. In the subsequent motion the 8 kg particle hits the ground and does not rebound. Find the time that elapses after the 8 kg particle hits the ground before the other two particles come to instantaneous rest. (You may assume this occurs before either particle reaches a pulley.)
    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 November Q2
5 marks Standard +0.3
2 Small smooth spheres \(A\) and \(B\), of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(A\) is moving towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving towards \(A\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the spheres collide, both \(A\) and \(B\) move in the same direction and the difference in the speeds of the spheres is \(2 \mathrm {~ms} ^ { - 1 }\). Find the loss of kinetic energy of the system due to the collision.
CAIE M1 2022 November Q3
9 marks Standard +0.3
3 A constant resistance of magnitude 1400 N acts on a car of mass 1250 kg .
  1. The car is moving along a straight level road at a constant speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, in kW , the rate at which the engine of the car is working.
  2. The car now travels at a constant speed up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.12\), with the engine working at 43.5 kW . Find this speed.
  3. On another occasion, the car pulls a trailer of mass 600 kg up the same hill. The system of the car and the trailer is modelled as particles connected by a light inextensible cable. The car's engine produces a driving force of 5000 N and the resistance to the motion of the trailer is 300 N . The resistance to the motion of the car remains 1400 N . Find the acceleration of the system and the tension in the cable. \includegraphics[max width=\textwidth, alt={}, center]{167f782c-3047-41f9-90a8-32ccdc19216d-06_378_631_255_757} A block of mass 8 kg is placed on a rough plane which is inclined at an angle of \(18 ^ { \circ }\) to the horizontal. The block is pulled up the plane by a light string that makes an angle of \(26 ^ { \circ }\) above a line of greatest slope. The tension in the string is \(T \mathrm {~N}\) (see diagram). The coefficient of friction between the block and plane is 0.65 .
  4. The acceleration of the block is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find \(T\).
  5. The block is initially at rest. Find the distance travelled by the block during the fourth second of motion.
CAIE M1 2022 November Q5
10 marks Standard +0.3
5 A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(- 20 \mathrm {~ms} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after leaving \(O\) is given by \(a = 12 - 2 t\).
  1. Sketch a velocity-time graph for \(0 \leqslant t \leqslant 12\), indicating the times when \(P\) is at rest.
  2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 12\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{167f782c-3047-41f9-90a8-32ccdc19216d-10_410_723_260_717} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{figure} Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.
  3. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane.
  4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{167f782c-3047-41f9-90a8-32ccdc19216d-11_412_899_276_589} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure} It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level.
    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 November Q1
3 marks Moderate -0.3
1 A cyclist is riding a bicycle along a straight horizontal road \(A B\) of length 50 m . The cyclist starts from rest at \(A\) and reaches a speed of \(6 \mathrm {~ms} ^ { - 1 }\) at \(B\). The cyclist produces a constant driving force of magnitude 100 N . There is a resistance force, and the work done against the resistance force from \(A\) to \(B\) is 3560 J . Find the total mass of the cyclist and bicycle.
CAIE M1 2022 November Q2
7 marks Standard +0.3
2 A particle \(P\) of mass 0.4 kg is in limiting equilibrium on a plane inclined at \(30 ^ { \circ }\) to the horizontal.
  1. Show that the coefficient of friction between the particle and the plane is \(\frac { 1 } { 3 } \sqrt { 3 }\).
    A force of magnitude 7.2 N is now applied to \(P\) directly up a line of greatest slope of the plane.
  2. Given that \(P\) starts from rest, find the time that it takes for \(P\) to move 1 m up the plane.
CAIE M1 2022 November Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{172e83d8-730c-4c18-aacb-ba2596886e41-04_412_601_260_772} A particle of mass 0.3 kg is held at rest by two light inextensible strings. One string is attached at an angle of \(60 ^ { \circ }\) to a horizontal ceiling. The other string is attached at an angle \(\alpha ^ { \circ }\) to a vertical wall (see diagram). The tension in the string attached to the ceiling is 4 N . Find the tension in the string which is attached to the wall and find the value of \(\alpha\).
CAIE M1 2022 November Q4
6 marks Standard +0.3
4 A car of mass 1200 kg is travelling along a straight horizontal road \(A B\). There is a constant resistance force of magnitude 500 N . When the car passes point \(A\), it has a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and an acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the power of the car's engine at the point \(A\).
    The car continues to work with this power as it travels from \(A\) to \(B\). The car takes 53 seconds to travel from \(A\) to \(B\) and the speed of the car at \(B\) is \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Show that the distance \(A B\) is 1362.6 m . \includegraphics[max width=\textwidth, alt={}, center]{172e83d8-730c-4c18-aacb-ba2596886e41-06_447_985_255_580} A block \(A\) of mass 80 kg is connected by a light, inextensible rope to a block \(B\) of mass 40 kg . The rope joining the two blocks is taut and is parallel to a line of greatest slope of a plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. A force of magnitude 500 N inclined at an angle of \(15 ^ { \circ }\) above the same line of greatest slope acts on \(A\) (see diagram). The blocks move up the plane and there is a resistance force of 50 N on \(B\), but no resistance force on \(A\).
  3. Find the acceleration of the blocks and the tension in the rope.
  4. Find the time that it takes for the blocks to reach a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest.
CAIE M1 2022 November Q6
9 marks Standard +0.3
6 Three particles \(A , B\) and \(C\) of masses \(0.3 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively lie at rest in a straight line on a smooth horizontal plane. The distance between \(B\) and \(C\) is \(2.1 \mathrm {~m} . A\) is projected directly towards \(B\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After \(A\) collides with \(B\) the speed of \(A\) is reduced to \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), still moving in the same direction.
  1. Show that the speed of \(B\) after the collision is \(1.05 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    After the collision between \(A\) and \(B , B\) moves directly towards \(C\). Particle \(B\) now collides with \(C\). After this collision, the two particles coalesce and have a combined speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find \(m\).
  3. Find the time that it takes, from the instant when \(B\) and \(C\) collide, until \(A\) collides with the combined particle.
CAIE M1 2022 November Q7
12 marks Standard +0.8
7 A particle \(P\) travels in a straight line, starting at rest from a point \(O\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is denoted by \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$\begin{array} { l l } a = 0.3 t ^ { \frac { 1 } { 2 } } & \text { for } 0 \leqslant t \leqslant 4 , \\ a = - k t ^ { - \frac { 3 } { 2 } } & \text { for } 4 < t \leqslant T , \end{array}$$ where \(k\) and \(T\) are constants.
  1. Find the velocity of \(P\) at \(t = 4\).
  2. It is given that there is no change in the velocity of \(P\) at \(t = 4\) and that the velocity of \(P\) at \(t = 16\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(k = 2.6\) and find an expression, in terms of \(t\), for the velocity of \(P\) for \(4 \leqslant t \leqslant T\).
  3. Given that \(P\) comes to instantaneous rest at \(t = T\), find the exact value of \(T\).
  4. Find the total distance travelled between \(t = 0\) and \(t = T\).
    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 November Q1
3 marks Easy -1.2
1 A particle \(P\) is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .
  1. Find \(u\).
  2. Find the greatest height of \(P\) above the ground.
CAIE M1 2022 November Q2
4 marks Moderate -0.3
2 A box of mass 5 kg is pulled at a constant speed of \(1.8 \mathrm {~ms} ^ { - 1 }\) for 15 s up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves along a line of greatest slope against a frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope.
  1. Find the change in gravitational potential energy of the box.
  2. Find the work done by the pulling force.