Questions M1 (1912 questions)

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CAIE M1 2017 June Q3
3 A particle \(A\) moves in a straight line with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds after \(A\) passes a point \(O\) on the line, a particle \(B\) passes through \(O\), moving along the line in the same direction as \(A\). Particle \(B\) has speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(O\) and has a constant deceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t \mathrm {~s}\) after \(B\) passes through \(O\).
  2. Find the distance between the particles when \(B\) comes to instantaneous rest.
  3. Find the minimum distance between the particles.
CAIE M1 2017 June Q4
4 A car of mass 1200 kg is moving on a straight road against a constant force of 850 N resisting the motion.
  1. On a part of the road that is horizontal, the car moves with a constant speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (a) Calculate, in kW , the power developed by the engine of the car.
    (b) Given that this power is suddenly increased by 6 kW , find the instantaneous acceleration of the car.
  2. On a part of the road that is inclined at \(\theta ^ { \circ }\) to the horizontal, the car moves up the hill at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the engine working at 80 kW . Find \(\theta\).
    \includegraphics[max width=\textwidth, alt={}, center]{39f94377-0dd0-4d1b-98de-815bd6e2c409-08_406_401_264_872} A particle of mass 0.12 kg is placed on a plane which is inclined at an angle of \(40 ^ { \circ }\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P \mathrm {~N}\) acting up the plane at an angle of \(30 ^ { \circ }\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is 0.32 . Find the set of possible values of \(P\).
CAIE M1 2017 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{39f94377-0dd0-4d1b-98de-815bd6e2c409-10_330_946_260_612} The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). A particle \(A\) of mass 0.3 kg rests on the horizontal surface and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the block. The other end of the string is attached to a particle \(B\) of mass 1.5 kg which rests on the sloping surface of the block. The system is released from rest with the string taut.
  1. Given that the block is smooth, find the acceleration of particle \(A\) and the tension in the string.
  2. It is given instead that the block is rough. The coefficient of friction between \(A\) and the block is \(\mu\) and the coefficient of friction between \(B\) and the block is also \(\mu\). In the first 3 seconds of the motion, \(A\) does not reach \(P\) and \(B\) does not reach the bottom of the sloping surface. The speed of the particles after 3 s is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the acceleration of particle \(A\) and the value of \(\mu\).
CAIE M1 2017 June Q1
1 A man pushes a wheelbarrow of mass 25 kg along a horizontal road with a constant force of magnitude 35 N at an angle of \(20 ^ { \circ }\) below the horizontal. There is a constant resistance to motion of 15 N . The wheelbarrow moves a distance of 12 m from rest.
  1. Find the work done by the man.
  2. Find the speed attained by the wheelbarrow after 12 m .
CAIE M1 2017 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-03_522_604_262_769} The four coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).
CAIE M1 2017 June Q3
3 A train travels between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and accelerates at a constant rate for \(T\) s until it reaches a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at this constant speed before decelerating at a constant rate, coming to rest at \(B\). The magnitude of the train's deceleration is twice the magnitude of its acceleration. The total time taken for the journey is 180 s .
  1. Sketch the velocity-time graph for the train's journey from \(A\) to \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-04_496_857_516_685}
  2. Find an expression, in terms of \(T\), for the length of time for which the train is travelling with constant speed.
  3. The distance from \(A\) to \(B\) is 3300 m . Find how far the train travels while it is decelerating.
CAIE M1 2017 June Q4
4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by \(v = ( 2 t - 5 ) ^ { 3 }\).
  1. Find the values of \(t\) when the acceleration of \(P\) is \(54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find an expression for the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\).
CAIE M1 2017 June Q5
5 A particle is projected vertically upwards from a point \(O\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds later a second particle is projected vertically upwards from \(O\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after the second particle is projected, the two particles collide.
  1. Find \(t\).
    \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-06_65_1569_488_328}
  2. Hence find the height above \(O\) at which the particles collide.
CAIE M1 2017 June Q6
6 A car of mass 1200 kg is travelling along a horizontal road.
  1. It is given that there is a constant resistance to motion.
    (a) The engine of the car is working at 16 kW while the car is travelling at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the resistance to motion.
    (b) The power is now increased to 22.5 kW . Find the acceleration of the car at the instant it is travelling at a speed of \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. It is given instead that the resistance to motion of the car is \(( 590 + 2 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels at a constant speed with the engine working at 16 kW . Find this speed.
CAIE M1 2017 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-10_336_803_258_671} Two particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and 4 kg respectively are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(A\) is on a rough fixed slope which is at an angle of \(30 ^ { \circ }\) to the horizontal ground. Particle \(B\) hangs vertically below the pulley and is 0.5 m above the ground (see diagram). The coefficient of friction between the slope and particle \(A\) is 0.2 .
  1. In the case where the system is in equilibrium with particle \(A\) on the point of moving directly up the slope, show that \(m = 5.94\), correct to 3 significant figures.
  2. In the case where \(m = 3\), the system is released from rest with the string taut. Find the total distance travelled by \(A\) before coming to instantaneous rest. You may assume that \(A\) does not reach the pulley.
CAIE M1 2018 June Q1
1 A particle \(P\) is projected vertically upwards with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 5 m above ground level. Find the time from projection until \(P\) reaches the ground.
\includegraphics[max width=\textwidth, alt={}, center]{610c1d96-3833-4a71-a71c-4cac13982332-03_424_677_260_735} The diagram shows three coplanar forces acting at the point \(O\). The magnitudes of the forces are 6 N , 8 N and 10 N . The angle between the 6 N force and the 8 N force is \(90 ^ { \circ }\). The forces are in equilibrium. Find the other angles between the forces.
CAIE M1 2018 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{610c1d96-3833-4a71-a71c-4cac13982332-04_355_533_260_806} A particle \(P\) of mass 8 kg is on a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. A force of magnitude 100 N , making an angle of \(\theta ^ { \circ }\) with a line of greatest slope and lying in the vertical plane containing the line of greatest slope, acts on \(P\) (see diagram).
  1. Given that \(P\) is in equilibrium, show that \(\theta = 66.4\), correct to 1 decimal place, and find the normal reaction between the plane and \(P\).
  2. Given instead that \(\theta = 30\), find the acceleration of \(P\).
CAIE M1 2018 June Q4
4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the displacement \(s \mathrm {~m}\) from \(O\) is given by \(s = t ^ { 3 } - 4 t ^ { 2 } + 4 t\) and the velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find an expression for \(v\) in terms of \(t\).
  2. Find the two values of \(t\) for which \(P\) is at instantaneous rest.
  3. Find the minimum velocity of \(P\).
CAIE M1 2018 June Q5
5 A sprinter runs a race of 200 m . His total time for running the race is 20 s . He starts from rest and accelerates uniformly for 6 s , reaching a speed of \(12 \mathrm {~ms} ^ { - 1 }\). He maintains this speed for the next 10 s , before decelerating uniformly to cross the finishing line with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the distance travelled by the sprinter in the first 16 s of the race. Hence sketch a displacementtime graph for the 20 s of the sprinter's race.
    \includegraphics[max width=\textwidth, alt={}, center]{610c1d96-3833-4a71-a71c-4cac13982332-08_1179_1363_1430_431}
  2. Find the value of \(V\).
CAIE M1 2018 June Q6
6 A car has mass 1250 kg .
  1. The car is moving along a straight level road at a constant speed of \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is subject to a constant resistance of magnitude 850 N . Find, in kW , the rate at which the engine of the car is working.
  2. The car travels at a constant speed up a hill and is subject to the same resistance as in part (i). The hill is inclined at an angle of \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta ^ { \circ } = 0.1\), and the engine is working at 63 kW . Find the speed of the car.
  3. The car descends the same hill with the engine of the car working at a constant rate of 20 kW . The resistance is not constant. The initial speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Eight seconds later the car has speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and has moved 176 m down the hill. Use an energy method to find the total work done against the resistance during the eight seconds.
CAIE M1 2018 June Q7
6 marks
7
\includegraphics[max width=\textwidth, alt={}, center]{610c1d96-3833-4a71-a71c-4cac13982332-12_365_831_264_657} The diagram shows a triangular block with sloping faces inclined to the horizontal at \(45 ^ { \circ }\) and \(30 ^ { \circ }\). Particle \(A\) of mass 0.8 kg lies on the face inclined at \(45 ^ { \circ }\) and particle \(B\) of mass 1.2 kg lies on the face inclined at \(30 ^ { \circ }\). The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the faces. The parts \(A P\) and \(B P\) of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.
  1. Given that both faces are smooth, find the speed of \(A\) after each particle has travelled a distance of 0.4 m .
  2. It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is \(\mu\). Find the value of \(\mu\) for which the system is in limiting equilibrium. [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 2018 June Q1
1 A man has mass 80 kg . He runs along a horizontal road against a constant resistance force of magnitude \(P \mathrm {~N}\). The total work done by the man in increasing his speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while running a distance of 60 metres is 1200 J . Find the value of \(P\).
CAIE M1 2018 June Q2
2 A train of mass 240000 kg travels up a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance of magnitude 18000 N acting on the train. At an instant when the speed of the train is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its deceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power of the engine of the train.
CAIE M1 2018 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{16640429-198d-4ea9-a2f6-6e2ef6ac1b4a-05_535_616_260_762} The three coplanar forces shown in the diagram have magnitudes \(3 \mathrm {~N} , 2 \mathrm {~N}\) and \(P \mathrm {~N}\). Given that the three forces are in equilibrium, find the values of \(\theta\) and \(P\).
CAIE M1 2018 June Q4
4 A particle \(P\) moves in a straight line \(A B C D\) with constant acceleration. The distances \(A B\) and \(B C\) are 100 m and 148 m respectively. The particle takes 4 s to travel from \(A\) to \(B\) and also takes 4 s to travel from \(B\) to \(C\).
  1. Show that the acceleration of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the speed of \(P\) at \(A\).
  2. \(P\) reaches \(D\) with a speed of \(61 \mathrm {~ms} ^ { - 1 }\). Find the distance \(C D\).
CAIE M1 2018 June Q5
5 A particle of mass 20 kg is on a rough plane inclined at an angle of \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude \(P \mathrm {~N}\) acting on the particle, in a direction parallel to a line of greatest slope of the plane. The greatest possible value of \(P\) is twice the least possible value of \(P\). Find the value of the coefficient of friction between the particle and the plane.
CAIE M1 2018 June Q6
6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of \(P\) is given by \(a = 6 - 0.24 t\). The particle comes to instantaneous rest at time \(t = 20\).
  1. Find the value of \(t\) at which the particle is again at instantaneous rest.
  2. Find the distance the particle travels between the times of instantaneous rest.
CAIE M1 2018 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{16640429-198d-4ea9-a2f6-6e2ef6ac1b4a-12_282_1106_264_523} As shown in the diagram, a particle \(A\) of mass 1.6 kg lies on a horizontal plane and a particle \(B\) of mass 2.4 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the inclined plane. The distance \(A P\) is 2.5 m and the distance of \(B\) from the bottom of the inclined plane is 1 m . There is a barrier at the bottom of the inclined plane preventing any further motion of \(B\). The part \(B P\) of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest with both parts of the string taut.
  1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
  2. It is given instead that the horizontal plane is rough and that the coefficient of friction between \(A\) and the horizontal plane is 0.2 . The inclined plane is smooth. Find the total distance travelled by \(A\).
    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 2018 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{03e325b9-171a-4f76-95cd-57dad3741caf-03_533_1351_260_397} The diagram shows the velocity-time graph for a train which travels from rest at one station to rest at the next station. The graph consists of three straight line segments. The distance between the two stations is 9040 m .
  1. Find the acceleration of the train during the first 40 s .
  2. Find the length of time for which the train is travelling at constant speed.
  3. Find the distance travelled by the train while it is decelerating.
CAIE M1 2018 June Q2
2 A small ball is projected vertically downwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) at a height of 7.2 m above horizontal ground. The ball hits the ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and rebounds vertically upwards with speed \(\frac { 1 } { 2 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The highest point the ball reaches after rebounding is \(B\). Find \(V\) and hence find the total time taken for the ball to reach the ground from \(A\) and rebound to \(B\).