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CAIE M1 2020 June Q3
6 marks Moderate -0.3
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
7 marks Moderate -0.3
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
9 marks Moderate -0.3
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
9 marks Standard +0.3
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
3 marks Moderate -0.3
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
6 marks Standard +0.8
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
7 marks Standard +0.3
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
8 marks Standard +0.3
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
9 marks Moderate -0.3
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
11 marks Standard +0.3
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
3 marks Moderate -0.8
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
6 marks Moderate -0.3
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
6 marks Standard +0.3
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.
CAIE M1 2021 June Q4
6 marks Standard +0.8
4 A particle of mass 12 kg is stationary on a rough plane inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A pulling force of magnitude \(P \mathrm {~N}\) acts at an angle of \(8 ^ { \circ }\) above a line of greatest slope of the plane. This force is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is 0.3 . Find the greatest possible value of \(P\).
CAIE M1 2021 June Q5
11 marks Standard +0.3
5 A car of mass 1250 kg is pulling a caravan of mass 800 kg along a straight road. The resistances to the motion of the car and caravan are 440 N and 280 N respectively. The car and caravan are connected by a light rigid tow-bar.
  1. The car and caravan move along a horizontal part of the road at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    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 and caravan and the tension in the tow-bar.
  2. The car and caravan now travel along a part of the road inclined at \(\sin ^ { - 1 } 0.06\) to the horizontal. The car and caravan travel up the incline at constant speed with the engine of the car working at 28 kW .
    1. Find this constant speed.
    2. Find the increase in the potential energy of the caravan in one minute.
CAIE M1 2021 June Q6
8 marks Challenging +1.2
6 A particle \(A\) is projected vertically upwards from level ground with an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant a particle \(B\) is released from rest 15 m vertically above \(A\). The mass of one of the particles is twice the mass of the other particle. During the subsequent motion \(A\) and \(B\) collide and coalesce to form particle \(C\). Find the difference between the two possible times at which \(C\) hits the ground. \(7 \quad\) A particle \(P\) moving in a straight line starts from rest at a point \(O\) and comes to rest 16 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(P\) is given by $$\begin{array} { l l } a = 6 + 4 t & 0 \leqslant t < 2 , \\ a = 14 & 2 \leqslant t < 4 , \\ a = 16 - 2 t & 4 \leqslant t \leqslant 16 . \end{array}$$ There is no sudden change in velocity at any instant.
  1. Find the values of \(t\) when the velocity of \(P\) is \(55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Complete the sketch of the velocity-time diagram. \includegraphics[max width=\textwidth, alt={}, center]{41e63d05-d109-47dc-80a6-927953e3e607-11_511_1054_351_584}
  3. Find the distance travelled by \(P\) when it is decelerating.
    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
4 marks Standard +0.3
1 Particles \(P\) of mass 0.4 kg and \(Q\) of mass 0.5 kg are free to move on a smooth horizontal plane. \(P\) and \(Q\) are moving directly towards each other with speeds \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. After \(P\) and \(Q\) collide, the speed of \(Q\) is twice the speed of \(P\). Find the two possible values of the speed of \(P\) after the collision.
CAIE M1 2021 June Q2
5 marks Standard +0.3
2 A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W . At an instant when her speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion is 20 N .
  1. Find the total mass of the cyclist and her bicycle.
    The cyclist comes to a straight hill inclined at an angle \(\theta\) above the horizontal. She ascends the hill at constant speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She continues to work at the same rate as before and the resistance force is unchanged.
  2. Find the value of \(\theta\).
CAIE M1 2021 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{ba29ddb2-3558-4be1-a8a8-134e27a70149-04_456_767_260_689} Four coplanar forces act at a point. The magnitudes of the forces are \(20 \mathrm {~N} , 30 \mathrm {~N} , 40 \mathrm {~N}\) and \(F \mathrm {~N}\). The directions of the forces are as shown in the diagram, where \(\sin \alpha ^ { \circ } = 0.28\) and \(\sin \beta ^ { \circ } = 0.6\). Given that the forces are in equilibrium, find \(F\) and \(\theta\).
CAIE M1 2021 June Q4
6 marks Moderate -0.3
4 A particle is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground. After 2 seconds, the height of the particle above the ground is 24 m .
  1. Show that \(u = 22\).
  2. The height of the particle above the ground is more than \(h \mathrm {~m}\) for a period of 3.6 s . Find \(h\).
CAIE M1 2021 June Q5
9 marks Standard +0.3
5 A car of mass 1400 kg is towing a trailer of mass 500 kg down a straight hill inclined at an angle of \(5 ^ { \circ }\) to the horizontal. The car and trailer are connected by a light rigid tow-bar. At the top of the hill the speed of the car and trailer is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at the bottom of the hill their speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. It is given that as the car and trailer descend the hill, the engine of the car does 150000 J of work, and there are no resistance forces. Find the length of the hill.
  2. It is given instead that there is a resistance force of 100 N on the trailer, the length of the hill is 200 m , and the acceleration of the car and trailer is constant. Find the tension in the tow-bar between the car and trailer.
CAIE M1 2021 June Q6
10 marks Moderate -0.3
6 A particle moves in a straight line and passes through the point \(A\) at time \(t = 0\). The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 2 t ^ { 2 } - 5 t + 3$$
  1. Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
  2. Sketch the velocity-time graph for the first 3 seconds of motion.
  3. Find the distance travelled between the two times when the particle is instantaneously at rest.
CAIE M1 2021 June Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ba29ddb2-3558-4be1-a8a8-134e27a70149-10_220_609_260_769} A particle \(P\) of mass 0.3 kg rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 7 } { 25 }\). A horizontal force of magnitude 4 N , acting in the vertical plane containing a line of greatest slope of the plane, is applied to \(P\) (see diagram). The particle is on the point of sliding up the plane.
  1. Show that the coefficient of friction between the particle and the plane is \(\frac { 3 } { 4 }\).
    The force acting horizontally is replaced by a force of magnitude 4 N acting up the plane parallel to a line of greatest slope.
  2. Find the acceleration of \(P\).
  3. Starting with \(P\) at rest, the force of 4 N parallel to the plane acts for 3 seconds and is then removed. Find the total distance travelled until \(P\) comes to instantaneous rest.
    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 June Q1
6 marks Easy -1.2
1 A car starts from rest and moves in a straight line with constant acceleration for a distance of 200 m , reaching a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then travels at this speed for 400 m , before decelerating uniformly to rest over a period of 5 s .
  1. Find the time for which the car is accelerating.
  2. Sketch the velocity-time graph for the motion of the car, showing the key points.
  3. Find the average speed of the car during its motion.
CAIE M1 2022 June Q2
5 marks Moderate -0.3
2 Two particles \(P\) and \(Q\), of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and \(P\) is vertically above \(Q\). A force of magnitude 10 N is applied to \(P\) vertically upwards. Find the acceleration of the particles and the tension in the string connecting them.