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CAIE M1 2020 November Q1
5 marks Easy -1.2
1 A particle \(B\) of mass 5 kg is at rest on a smooth horizontal table. A particle \(A\) of mass 2.5 kg moves on the table with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and collides directly with \(B\). In the collision the two particles coalesce.
  1. Find the speed of the combined particle after the collision.
  2. Find the loss of kinetic energy of the system due to the collision.
CAIE M1 2020 November Q2
5 marks Moderate -0.8
2 A car of mass 1400 kg is moving along a straight horizontal road against a resistance of magnitude 350 N .
  1. Find, in kW , the rate at which the engine of the car is working when it is travelling at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the acceleration of the car when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at 15 kW .
CAIE M1 2020 November Q3
6 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-04_378_969_258_587} Coplanar forces of magnitudes \(8 \mathrm {~N} , 12 \mathrm {~N} , 10 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\).
CAIE M1 2020 November Q4
6 marks Standard +0.3
4 A particle \(P\) moves in a straight line. It starts from rest at a point \(O\) on the line and at time \(t \mathrm {~s}\) after leaving \(O\) it has acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 18\). Find the distance \(P\) moves before it comes to instantaneous rest. \includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-06_540_606_258_767} Two particles of masses 0.8 kg and 0.2 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The system is released from rest with both particles 0.5 m above a horizontal floor (see diagram). In the subsequent motion the 0.2 kg particle does not reach the pulley.
  1. Show that the magnitude of the acceleration of the particles is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. When the 0.8 kg particle reaches the floor it comes to rest. Find the greatest height of the 0.2 kg particle above the floor.
CAIE M1 2020 November Q6
9 marks Standard +0.3
6 A car of mass 1500 kg is pulling a trailer of mass 750 kg up a straight hill of length 800 m inclined at an angle of \(\sin ^ { - 1 } 0.08\) to the horizontal. The resistances to the motion of the car and trailer are 400 N and 200 N respectively. The car and trailer are connected by a light rigid tow-bar. The car and trailer have speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill and \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill.
  1. Use an energy method to find the constant driving force as the car and trailer travel up the hill.
    [0pt] [5]
    ..................................................................................................................................
    After reaching the top of the hill the system consisting of the car and trailer travels along a straight level road. The driving force of the car's engine is 2400 N and the resistances to motion are unchanged.
  2. Find the acceleration of the system and the tension in the tow-bar. \includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-10_440_738_262_699} Three points \(A , B\) and \(C\) lie on a line of greatest slope of a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal, with \(A B = 1 \mathrm {~m}\) and \(B C = 1 \mathrm {~m}\), as shown in the diagram. A particle of mass 0.2 kg is released from rest at \(A\) and slides down the plane. The part of the plane from \(A\) to \(B\) is smooth. The part of the plane from \(B\) to \(C\) is rough, with coefficient of friction \(\mu\) between the plane and the particle.
  3. Given that \(\mu = \frac { 1 } { 2 } \sqrt { 3 }\), find the speed of the particle at \(C\).
  4. Given instead that the particle comes to rest at \(C\), find the exact value of \(\mu\).
    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 2020 November Q1
3 marks Easy -1.3
1 Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down the momentum of \(P\).
  2. After the collision \(P\) continues to move in the same direction with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.
CAIE M1 2020 November Q2
5 marks Moderate -0.3
2 A car of mass 1800 kg is travelling along a straight horizontal road. The power of the car's engine is constant. There is a constant resistance to motion of 650 N .
  1. Find the power of the car's engine, given that the car's acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the steady speed which the car can maintain with the engine working at this power.
CAIE M1 2020 November Q3
5 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-05_446_851_260_646} A block of mass \(m \mathrm {~kg}\) is held in equilibrium below a horizontal ceiling by two strings, as shown in the diagram. One of the strings is inclined at \(45 ^ { \circ }\) to the horizontal and the tension in this string is \(T \mathrm {~N}\). The other string is inclined at \(60 ^ { \circ }\) to the horizontal and the tension in this string is 20 N . Find \(T\) and \(m\).
CAIE M1 2020 November Q4
5 marks Moderate -0.5
4 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-06_602_1203_260_470} The diagram shows a velocity-time graph which models the motion of a car. The graph consists of four straight line segments. The car accelerates at a constant rate of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(T \mathrm {~s}\). It then decelerates at a constant rate for 5 seconds before travelling at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 27.5 s . The car then decelerates to rest at a constant rate over a period of 5 s .
  1. Find \(T\).
  2. Given that the distance travelled up to the point at which the car begins to move with constant speed is one third of the total distance travelled, find \(V\).
CAIE M1 2020 November Q5
8 marks Standard +0.3
5 A particle is projected vertically upwards with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) alongside a building of height \(h \mathrm {~m}\).
  1. Given that the particle is above the level of the top of the building for 4 s , find \(h\).
  2. One second after the first particle is projected, a second particle is projected vertically upwards from the top of the building with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Denoting the time after projection of the first particle by \(t \mathrm {~s}\), find the value of \(t\) for which the two particles are at the same height above the ground.
CAIE M1 2020 November Q6
8 marks Standard +0.3
6 A block of mass 5 kg is placed on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is \(\mu\).
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-10_424_709_392_760} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{figure} When a force of magnitude 40 N is applied to the block, acting up the plane parallel to a line of greatest slope, the block begins to slide up the plane (see Fig. 6.1). Show that \(\mu < \frac { 1 } { 5 } \sqrt { 3 }\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-11_422_727_264_749} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure} When a force of magnitude 40 N is applied horizontally, in a vertical plane containing a line of greatest slope, the block does not move (see Fig. 6.2). Show that, correct to 3 decimal places, the least possible value of \(\mu\) is 0.152 .
CAIE M1 2020 November Q7
7 marks Moderate -0.3
7 A particle \(P\) moves in a straight line, starting from a point \(O\) with velocity \(1.72 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle, \(t \mathrm {~s}\) after leaving \(O\), is given by \(a = 0.1 t ^ { \frac { 3 } { 2 } }\).
  1. Find the value of \(t\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the displacement of \(P\) from \(O\) when \(t = 2\), giving your answer correct to 2 decimal places. [3]
CAIE M1 2020 November Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-14_388_1216_264_461} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to a horizontal plane and to the top of an inclined plane. The particles are initially at rest with \(A\) on the horizontal plane and \(B\) on the inclined plane, which makes an angle of \(30 ^ { \circ }\) with the horizontal. The string is taut and \(B\) can move on a line of greatest slope of the inclined plane. A force of magnitude 3.5 N is applied to \(B\) acting down the plane (see diagram).
  1. Given that both planes are smooth, find the tension in the string and the acceleration of \(B\).
  2. It is given instead that the two planes are rough. When each particle has moved a distance of 0.6 m from rest, the total amount of work done against friction is 1.1 J . Use an energy method to find the speed of \(B\) when it has moved this distance down the plane. [You should assume that the string is sufficiently long so that \(A\) does not hit the pulley when it moves 0.6 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 2020 November Q1
3 marks Easy -1.2
1 A particle \(P\) is projected vertically upwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .
  1. Find \(v\).
  2. Find the greatest height of \(P\) above the ground.
CAIE M1 2020 November Q2
4 marks Moderate -0.8
2 A box of mass 5 kg is pulled at a constant speed a distance of 15 m 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 work done against friction.
  2. Find the change in gravitational potential energy of the box.
  3. Find the work done by the pulling force.
CAIE M1 2020 November Q3
6 marks Moderate -0.8
3 A string is attached to a block of mass 4 kg which rests in limiting equilibrium on a rough horizontal table. The string makes an angle of \(24 ^ { \circ }\) above the horizontal and the tension in the string is 30 N .
  1. Draw a diagram showing all the forces acting on the block.
  2. Find the coefficient of friction between the block and the table.
CAIE M1 2020 November Q4
6 marks Standard +0.3
4 Two small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and \(m \mathrm {~kg}\) respectively, lie on a smooth horizontal plane. Initially, sphere \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\). After the collision \(A\) moves with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the two possible values of the loss of kinetic energy due to the collision.
CAIE M1 2020 November Q5
10 marks Moderate -0.8
5 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t ^ { 2 } - 20 t + 21\).
  1. Find the values of \(t\) for which \(P\) is at instantaneous rest.
  2. Find the initial acceleration of \(P\).
  3. Find the minimum velocity of \(P\).
  4. Find the distance travelled by \(P\) during the time when its velocity is negative.
CAIE M1 2020 November Q6
10 marks Moderate -0.3
6 A car of mass 1600 kg is pulling a caravan of mass 800 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 400 N and 250 N respectively.
  1. The car and caravan are travelling along a straight horizontal road.
    1. Given that the car and caravan have a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the power of the car's engine.
    2. The engine's power is now suddenly increased to 39 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
  2. The car and caravan now travel up a straight hill, inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 32.5 kW . Find \(v\).
CAIE M1 2020 November Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ac4bb5a0-c7c0-4e1d-9e76-64f92ae28066-10_214_1461_255_342} As shown in the diagram, particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle \(A\) is on plane \(P\), which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. Particle \(B\) is on plane \(Q\), which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes.
  1. It is given that plane \(P\) is smooth, plane \(Q\) is rough, and the particles are in limiting equilibrium. Find the coefficient of friction between particle \(B\) and plane \(Q\).
  2. It is given instead that both planes are smooth and that the particles are released from rest at the same horizontal level. Find the time taken until the difference in the vertical height of the particles is 1 m . [You should assume that this occurs before \(A\) reaches the pulley or \(B\) reaches the bottom of plane \(Q\).] [6]
    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
3 marks Moderate -0.3
1 A bus moves from rest with constant acceleration for 12 s . It then moves with constant speed for 30 s before decelerating uniformly to rest in a further 6 s . The total distance travelled is 585 m .
  1. Find the constant speed of the bus.
  2. Find the magnitude of the deceleration.
CAIE M1 2021 November Q2
5 marks Moderate -0.8
2 Two small smooth spheres \(A\) and \(B\), of equal radii and of masses km kg and \(m \mathrm {~kg}\) respectively, where \(k > 1\), are free to move on a smooth horizontal plane. \(A\) is moving towards \(B\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving towards \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision \(A\) and \(B\) coalesce and move with speed \(4 \mathrm {~ms} ^ { - 1 }\).
  1. Find \(k\).
  2. Find, in terms of \(m\), the loss of kinetic energy due to the collision.
CAIE M1 2021 November Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-04_519_616_260_762} Coplanar forces of magnitudes \(24 \mathrm {~N} , P \mathrm {~N} , 20 \mathrm {~N}\) and 36 N act at a point in the directions shown in the diagram. The system is in equilibrium. Given that \(\sin \alpha = \frac { 3 } { 5 }\), find the values of \(P\) and \(\theta\).
CAIE M1 2021 November Q4
6 marks Moderate -0.3
4 A particle of mass 12 kg is stationary on a rough plane inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A force of magnitude \(P \mathrm {~N}\) acting parallel to a line of greatest slope of the plane is used to prevent the particle sliding down the plane. The coefficient of friction between the particle and the plane is 0.35 .
  1. Draw a sketch showing the forces acting on the particle.
  2. Find the least possible value of \(P\).
CAIE M1 2021 November Q5
11 marks Standard +0.3
5 A car of mass 1600 kg travels at constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle of \(\sin ^ { - 1 } 0.12\) to the horizontal.
  1. Find the change in potential energy of the car in 30 s .
  2. Given that the total work done by the engine of the car in this time is 1960 kJ , find the constant force resisting the motion.
  3. Calculate, in kW , the power developed by the engine of the car.
  4. Given that this power is suddenly decreased by \(15 \%\), find the instantaneous deceleration of the car.