Questions M1 (1912 questions)

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CAIE M1 2020 June Q2
2 A car of mass 1800 kg is towing a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There are constant resistance forces of 250 N on the car and 100 N on the trailer.
  1. Find the tension in the tow-bar.
  2. Find the power of the engine of the car at the instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2020 June Q3
3 A particle \(P\) is projected vertically upwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 2.8 m above horizontal ground.
  1. Find the greatest height above the ground reached by \(P\).
  2. Find the length of time for which \(P\) is at a height of more than 3.6 m above the ground.
    The diagram shows a ring of mass 0.1 kg threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is 0.8 . A force of magnitude \(T \mathrm {~N}\) acts on the ring in a direction at \(30 ^ { \circ }\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest.
  3. Find the greatest value of \(T\) for which the ring remains at rest.
  4. Find the acceleration of the ring when \(T = 3\).
CAIE M1 2020 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{c175972b-e298-4c86-bbef-3b05c6aca76f-08_412_339_260_897} A child of mass 35 kg is swinging on a rope. The child is modelled as a particle \(P\) and the rope is modelled as a light inextensible string of length 4 m . Initially \(P\) is held at an angle of \(45 ^ { \circ }\) to the vertical (see diagram).
  1. Given that there is no resistance force, find the speed of \(P\) when it has travelled half way along the circular arc from its initial position to its lowest point.
  2. It is given instead that there is a resistance force. The work done against the resistance force as \(P\) travels from its initial position to its lowest point is \(X \mathrm {~J}\). The speed of \(P\) at its lowest point is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(X\).
CAIE M1 2020 June Q6
6 A particle moves in a straight line \(A B\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle \(t \mathrm {~s}\) after leaving \(A\) is given by \(v = k \left( t ^ { 2 } - 10 t + 21 \right)\), where \(k\) is a constant. The displacement of the particle from \(A\), in the direction towards \(B\), is 2.85 m when \(t = 3\) and is 2.4 m when \(t = 6\).
  1. Find the value of \(k\). Hence find an expression, in terms of \(t\), for the displacement of the particle from \(A\).
  2. Find the displacement of the particle from \(A\) when its velocity is a minimum.
CAIE M1 2020 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{c175972b-e298-4c86-bbef-3b05c6aca76f-12_399_1121_262_511} A particle \(P\) of mass 0.3 kg , lying on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of 2.5 m and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass 0.2 kg lies at rest on the horizontal plane 1.5 m from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\).
  1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.
  2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the coefficient of friction between \(P\) and the horizontal plane.
    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 June Q1
2 marks
1 A tram starts from rest and moves with uniform acceleration for 20 s . The tram then travels at a constant speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for 170 s before being brought to rest with a uniform deceleration of magnitude twice that of the acceleration. The total distance travelled by the tram is 2.775 km .
  1. Sketch a velocity-time graph for the motion, stating the total time for which the tram is moving.
    [0pt] [2]
  2. Find \(V\).
  3. Find the magnitude of the acceleration.
    \includegraphics[max width=\textwidth, alt={}, center]{e11bebff-1c09-4576-9b9b-a1678ea2b226-03_625_627_260_758} Coplanar forces of magnitudes \(20 \mathrm {~N} , P \mathrm {~N} , 3 P \mathrm {~N}\) and \(4 P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\).
CAIE M1 2020 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{e11bebff-1c09-4576-9b9b-a1678ea2b226-04_397_759_264_694} A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal by a force of magnitude \(T \mathrm {~N}\) making an angle of \(60 ^ { \circ }\) with a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.3 . Find the greatest and least possible values of \(T\).
CAIE M1 2020 June Q4
4 Small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the spheres collide \(A\) continues to move in the same direction but with half the speed of \(B\).
  1. Find the speed of \(B\) after the collision.
    A third small smooth sphere \(C\), of mass 1 kg and with the same radius as \(A\) and \(B\), is at rest on the plane. \(B\) now collides directly with \(C\). After this collision \(B\) continues to move in the same direction but with one third the speed of \(C\).
  2. Show that there is another collision between \(A\) and \(B\).
  3. \(\quad A\) and \(B\) coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions.
CAIE M1 2020 June Q5
5 A car of mass 1250 kg is moving on a straight road.
  1. On a horizontal section of the road, the car has a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and there is a constant force of 750 N resisting the motion.
    1. Calculate, in kW , the power developed by the engine of the car.
    2. Given that this power is suddenly decreased by 8 kW , find the instantaneous deceleration of the car.
  2. On a section of the road inclined at \(\sin ^ { - 1 } 0.096\) to the horizontal, the resistance to the motion of the car is \(( 1000 + 8 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels up this section of the road at constant speed with the engine working at 60 kW . Find this constant speed.
CAIE M1 2020 June Q6
6 A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 2 t + 1 & \text { for } 0 \leqslant t \leqslant 5 ,
v = 36 - t ^ { 2 } & \text { for } 5 \leqslant t \leqslant 7 ,
v = 2 t - 27 & \text { for } 7 \leqslant t \leqslant 13.5 . \end{array}$$
  1. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 13.5\).
  2. Find the acceleration at the instant when \(t = 6\).
  3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 13.5\).
    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 June Q1
1 Particles \(P\) of mass \(m \mathrm {~kg}\) and \(Q\) of mass 0.2 kg are free to move on a smooth horizontal plane. \(P\) is projected at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After the collision \(P\) and \(Q\) move in opposite directions with speeds of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find \(m\).
CAIE M1 2020 June Q2
2 A minibus of mass 4000 kg is travelling along a straight horizontal road. The resistance to motion is 900 N .
  1. Find the driving force when the acceleration of the minibus is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the power required for the minibus to maintain a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2020 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-04_586_1003_260_571} Four coplanar forces of magnitudes \(40 \mathrm {~N} , 20 \mathrm {~N} , 50 \mathrm {~N}\) and \(F \mathrm {~N}\) act at a point in the directions shown in the diagram. The four forces are in equilibrium. Find \(F\) and \(\alpha\).
CAIE M1 2020 June Q4
4 A car starts from rest and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 50 m . The car then travels with constant velocity for 500 m for a period of 25 s , before decelerating to rest. The magnitude of this deceleration is \(2 a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Sketch the velocity-time graph for the motion of the car.
    \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-05_533_1155_534_534}
  2. Find the value of \(a\).
  3. Find the total time for which the car is in motion.
CAIE M1 2020 June Q5
5 A block \(B\) of mass 4 kg is pushed up a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal by a force applied to \(B\), acting in the direction of motion of \(B\). The block passes through points \(P\) and \(Q\) with speeds \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(P\) and \(Q\) are 10 m apart with \(P\) below the level of \(Q\).
  1. Find the decrease in kinetic energy of the block as it moves from \(P\) to \(Q\).
  2. Hence find the work done by the force pushing the block up the slope as the block moves from \(P\) to \(Q\).
  3. At the instant the block reaches \(Q\), the force pushing the block up the slope is removed. Find the time taken, after this instant, for the block to return to \(P\).
CAIE M1 2020 June Q6
6 A particle travels in a straight line \(P Q\). The velocity of the particle \(t \mathrm {~s}\) after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 4.5 + 4 t - 0.5 t ^ { 2 }$$
  1. Find the velocity of the particle at the instant when its acceleration is zero.
    The particle comes to instantaneous rest at \(Q\).
  2. Find the distance \(P Q\).
    \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-10_625_780_260_744} Two particles \(A\) and \(B\), of masses \(3 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle \(\theta\) to the horizontal. \(A\) lies on the plane and \(B\) hangs vertically, 0.8 m above the floor, which is horizontal. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially \(A\) and \(B\) are at rest.
  3. Given that the plane is smooth, find the value of \(\theta\) for which \(A\) remains at rest.
    It is given instead that the plane is rough, \(\theta = 30 ^ { \circ }\) and the acceleration of \(A\) up the plane is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  4. Show that the coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 10 } \sqrt { 3 }\).
  5. When \(B\) reaches the floor it comes to rest. Find the length of time after \(B\) reaches the floor for which \(A\) is moving up the plane. [You may assume that \(A\) 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 June Q1
1 A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass 50 kg up a line of greatest slope of a plane inclined at \(60 ^ { \circ }\) to the horizontal. The winch pulls the load a distance of 5 m up the plane at constant speed. There is a constant resistance to motion of 100 N . Find the work done by the winch.
\includegraphics[max width=\textwidth, alt={}, center]{f14ab5b6-f9eb-46a5-b5a6-3d34433313c6-03_585_611_258_765} Two particles \(A\) and \(B\) have masses \(m \mathrm {~kg}\) and 0.1 kg respectively, where \(m > 0.1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.9 m above horizontal ground (see diagram). The system is released from rest, and while both particles are in motion the tension in the string is 1.5 N . Particle \(B\) does not reach the pulley.
  1. Find \(m\).
  2. Find the speed at which \(A\) reaches the ground.
CAIE M1 2021 June Q3
3 Three particles \(P , Q\) and \(R\), of masses \(0.1 \mathrm {~kg} , 0.2 \mathrm {~kg}\) and 0.5 kg respectively, are at rest in a straight line on a smooth horizontal plane. Particle \(P\) is projected towards \(Q\) at a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After \(P\) and \(Q\) collide, \(P\) rebounds with speed \(1 \mathrm {~ms} ^ { - 1 }\).
  1. Find the speed of \(Q\) immediately after the collision with \(P\).
    \(Q\) now collides with \(R\). Immediately after the collision with \(Q , R\) begins to move with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Given that there is no subsequent collision between \(P\) and \(Q\), find the greatest possible value of \(V\).
CAIE M1 2021 June Q4
4 Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with constant acceleration, starting from rest at point \(A\). Isabella accelerates for 5 s at a constant rate \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). She then travels at the constant speed she has reached for 10 s , before decelerating to rest at a constant rate over a period of 5 s . Maria accelerates at a constant rate, reaching a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a distance of 27.5 m . She then maintains this speed for a period of 10 s , before decelerating to rest at a constant rate over a period of 5 s .
  1. Given that \(a = 1.1\), find which cyclist travels further.
  2. Find the value of \(a\) for which the two cyclists travel the same distance.
CAIE M1 2021 June Q5
5 A particle moving in a straight line starts from rest at a point \(A\) and comes instantaneously to rest at a point \(B\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(A\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 6 t ^ { \frac { 1 } { 2 } } - 2 t$$
  1. Find the value of \(t\) at point \(B\).
  2. Find the distance travelled from \(A\) to the point at which the acceleration of the particle is again zero.
CAIE M1 2021 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{f14ab5b6-f9eb-46a5-b5a6-3d34433313c6-08_615_693_260_726} Three coplanar forces of magnitudes \(10 \mathrm {~N} , 25 \mathrm {~N}\) and 20 N act at a point \(O\) in the directions shown in the diagram.
  1. Given that the component of the resultant force in the \(x\)-direction is zero, find \(\alpha\), and hence find the magnitude of the resultant force.
  2. Given instead that \(\alpha = 45\), find the magnitude and direction of the resultant of the three forces.
CAIE M1 2021 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{f14ab5b6-f9eb-46a5-b5a6-3d34433313c6-10_326_938_262_609} A slide in a playground descends at a constant angle of \(30 ^ { \circ }\) for 2.5 m . It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg , modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.
  1. It is given that the sloping section of the slide is smooth.
    1. Find the speed of the child when she reaches the bottom of the sloping section.
    2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest.
  2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide.
    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 June Q1
1 A particle of mass 0.6 kg is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. Use an energy method to find the speed of the particle after it has moved 15 m down the plane.
CAIE M1 2021 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{41e63d05-d109-47dc-80a6-927953e3e607-03_659_655_258_744} Coplanar forces of magnitudes \(34 \mathrm {~N} , 30 \mathrm {~N}\) and 26 N act at a point in the directions shown in the diagram. Given that \(\sin \alpha = \frac { 5 } { 13 }\) and \(\sin \theta = \frac { 8 } { 17 }\), find the magnitude and direction of the resultant of the three forces.
CAIE M1 2021 June Q3
3 A ring of mass 0.3 kg is threaded on a horizontal rough rod. The coefficient of friction between the ring and the rod is 0.8 . A force of magnitude 8 N acts on the ring. This force acts at an angle of \(10 ^ { \circ }\) above the horizontal in the vertical plane containing the rod. Find the time taken for the ring to move, from rest, 0.6 m along the rod.