Questions M1 (1912 questions)

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CAIE M1 2016 June Q6
6 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 a velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 6 t ^ { 2 } - 30 t + 24\).
  1. Find the set of values of \(t\) for which the acceleration of the particle is negative.
  2. Find the distance between the two positions at which \(P\) is at instantaneous rest.
  3. Find the two positive values of \(t\) at which \(P\) passes through \(O\).
CAIE M1 2016 June Q7
7 A particle of mass 30 kg is on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. Starting from rest, the particle is pulled up the plane by a force of magnitude 200 N acting parallel to a line of greatest slope.
  1. Given that the plane is smooth, find
    (a) the acceleration of the particle,
    (b) the change in kinetic energy after the particle has moved 12 m up the plane.
  2. It is given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.12 .
    (a) Find the acceleration of the particle.
    (b) The direction of the force of magnitude 200 N is changed, and the force now acts at an angle of \(10 ^ { \circ }\) above the line of greatest slope. Find the acceleration of the particle.
CAIE M1 2016 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_373_591_260_776} Coplanar forces of magnitudes \(7 \mathrm {~N} , 6 \mathrm {~N}\) and 8 N act at a point in the directions shown in the diagram. Given that \(\sin \alpha = \frac { 3 } { 5 }\), find the magnitude and direction of the resultant of the three forces.
CAIE M1 2016 June Q2
2 A particle \(P\) moves in a straight line, starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = 4 t ^ { 2 } - 8 t + 3\).
  1. Find the two values of \(t\) at which \(P\) is at instantaneous rest.
  2. Find the distance travelled by \(P\) between these two times.
CAIE M1 2016 June Q3
3 A particle of mass 8 kg is projected with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). The motion of the particle is resisted by a constant frictional force of magnitude 15 N . The particle comes to instantaneous rest after travelling a distance \(x \mathrm {~m}\) up the plane.
  1. Express the change in gravitational potential energy of the particle in terms of \(x\).
  2. Use an energy method to find \(x\).
CAIE M1 2016 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_522_959_1692_593} A sprinter runs a race of 400 m . His total time for running the race is 52 s . The diagram shows the velocity-time graph for the motion of the sprinter. He starts from rest and accelerates uniformly to a speed of \(8.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 s . The sprinter maintains a speed of \(8.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 36 s , and he then decelerates uniformly to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race.
  1. Calculate the distance covered by the sprinter in the first 42 s of the race.
  2. Show that \(V = 7.84\).
  3. Calculate the deceleration of the sprinter in the last 10 s of the race.
CAIE M1 2016 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_394_531_260_806} A block of mass 2.5 kg is placed on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The block is kept in equilibrium by a light string making an angle of \(20 ^ { \circ }\) above a line of greatest slope. The tension in the string is \(T \mathrm {~N}\), as shown in the diagram. The coefficient of friction between the block and plane is \(\frac { 1 } { 4 }\). The block is in limiting equilibrium and is about to move up the plane. Find the value of \(T\).
CAIE M1 2016 June Q6
6 A car of mass 1100 kg is moving on a road against a constant force of 1550 N resisting the motion.
  1. The car moves along a straight horizontal road at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (a) Calculate, in kW , the power developed by the engine of the car.
    (b) Given that this power is suddenly decreased by 22 kW , find the instantaneous deceleration of the car.
  2. The car now travels at constant speed up a straight road inclined at \(8 ^ { \circ }\) to the horizontal, with the engine working at 80 kW . Assuming the resistance force remains the same, find this constant speed.
CAIE M1 2016 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_378_1001_1672_573} A particle \(A\) of mass 1.6 kg rests on a horizontal table 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 table. The other end of the string is attached to a particle \(B\) of mass 2.4 kg which hangs freely below the pulley. The system is released from rest with the string taut and with \(B\) at a height of 0.5 m above the ground, as shown in the diagram. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground.
  1. Given that the table is smooth, find the time taken by \(B\) to reach the ground.
  2. Given instead that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac { 3 } { 8 }\), find the total distance travelled by \(A\). You may assume that \(A\) does not reach the pulley.
CAIE M1 2016 June Q1
1 A particle of mass 8 kg is pulled at a constant speed a distance of 20 m up a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal by a force acting along a line of greatest slope.
  1. Find the change in gravitational potential energy of the particle.
  2. The total work done against gravity and friction is 1146 J . Find the frictional force acting on the particle.
CAIE M1 2016 June Q2
2 Alan starts walking from a point \(O\), at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along a horizontal path. Ben walks along the same path, also starting from \(O\). Ben starts from rest 5 s after Alan and accelerates at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . Ben then continues to walk at a constant speed until he is at the same point, \(P\), as Alan.
  1. Find how far Ben has travelled when he has been walking for 5 s and find his speed at this instant.
  2. Find the distance \(O P\).
CAIE M1 2016 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{aaf655c6-47f0-4f17-9a57-58aaf48728df-2_586_611_1171_767} The coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).
CAIE M1 2016 June Q4
4 A particle of mass 15 kg is stationary on a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.2 . A force of magnitude \(X \mathrm {~N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. Show that the least possible value of \(X\) is 23.1 , correct to 3 significant figures, and find the greatest possible value of \(X\).
CAIE M1 2016 June Q5
5 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N .
  1. Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate of 20 kW .
  2. The car is travelling at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). Find the power of the car's engine.
  3. The car descends the same hill with the engine working at \(80 \%\) of the power found in part (ii). Find the acceleration of the car at an instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2016 June Q6
6 Two particles of masses 1.3 kg and 0.7 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held at the same vertical height with the string taut. The distance of each particle above a horizontal plane is 2 m , and the distance of each particle below the pulley is 4 m . The particles are released from rest.
  1. Find
    (a) the tension in the string before the particle of mass 1.3 kg reaches the plane,
    (b) the time taken for the particle of mass 1.3 kg to reach the plane.
  2. Find the greatest height of the particle of mass 0.7 kg above the plane.
CAIE M1 2016 June Q7
7 A particle \(P\) moves in a straight line. At time \(t \mathrm {~s}\), the displacement of \(P\) from \(O\) is \(s \mathrm {~m}\) and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 2\). When \(t = 1 , s = 7\) and when \(t = 3 , s = 29\).
  1. Find the set of values of \(t\) for which the particle is decelerating.
  2. Find \(s\) in terms of \(t\).
  3. Find the time when the velocity of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2017 June Q1
1 A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done against air resistance.
CAIE M1 2017 June Q2
2 A particle of mass 0.8 kg is projected with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a rough plane inclined at an angle of \(10 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.4 .
  1. Find the acceleration of the particle.
  2. Find the distance the particle moves up the plane before coming to rest.
CAIE M1 2017 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-04_442_584_255_778} Two light inextensible strings are attached to a particle of weight 25 N . The strings pass over two smooth fixed pulleys and have particles of weights \(A \mathrm {~N}\) and \(B \mathrm {~N}\) hanging vertically at their ends. The sloping parts of the strings make angles of \(30 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the vertical (see diagram). The system is in equilibrium. Find the values of \(A\) and \(B\).
CAIE M1 2017 June Q4
4 A car of mass 800 kg is moving up a hill inclined at \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta = 0.15\). The initial speed of the car is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Twelve seconds later the car has travelled 120 m up the hill and has speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the change in the kinetic energy and the change in gravitational potential energy of the car.
  2. The engine of the car is working at a constant rate of 32 kW . Find the total work done against the resistive forces during the twelve seconds.
CAIE M1 2017 June Q5
5 A particle \(P\) moves in a straight line \(A B C D\) with constant deceleration. The velocities of \(P\) at \(A , B\) and \(C\) are \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  1. Find the ratio of distances \(A B : B C\).
  2. The particle comes to rest at \(D\). Given that the distance \(A D\) is 80 m , find the distance \(B C\).
CAIE M1 2017 June Q6
6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = q t + r t ^ { 2 }\), where \(q\) and \(r\) are constants. The particle has velocity \(4 \mathrm {~ms} ^ { - 1 }\) when \(t = 1\) and when \(t = 2\).
  1. Show that, when \(t = 0.5\), the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    …………………………………………………………………………………………......................... .
  2. Find the values of \(t\) when \(P\) is at instantaneous rest.
  3. The particle is at \(O\) when \(t = 3\). Find the distance of \(P\) from \(O\) when \(t = 0\).
CAIE M1 2017 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-10_374_762_259_688} As shown in the diagram, a particle \(A\) of mass 0.8 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal and a particle \(B\) of mass 1.2 kg lies on a plane inclined at an angle of \(60 ^ { \circ }\) 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 planes. The parts \(A P\) and \(B P\) of the string are parallel to lines of greatest slope of the respective planes. 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 both planes are rough, with the same coefficient of friction, \(\mu\), for both particles. Find the value of \(\mu\) for which the system is in limiting equilibrium.
CAIE M1 2017 June Q1
3 marks
1 One end of a light inextensible string is attached to a block. The string makes an angle of \(\theta ^ { \circ }\) with the horizontal. The tension in the string is 20 N . The string pulls the block along a horizontal surface at a constant speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 12 s . The work done by the tension in the string is 50 J . Find \(\theta\). [3]
CAIE M1 2017 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{39f94377-0dd0-4d1b-98de-815bd6e2c409-02_438_565_1347_790} The diagram shows a wire \(A B C D\) consisting of a straight part \(A B\) of length 5 m and a part \(B C D\) in the shape of a semicircle of radius 6 m and centre \(O\). The diameter \(B D\) of the semicircle is horizontal and \(A B\) is vertical. A small ring is threaded onto the wire and slides along the wire. The ring starts from rest at \(A\). The part \(A B\) of the wire is rough, and the ring accelerates at a constant rate of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) between \(A\) and \(B\).
  1. Show that the speed of the ring as it reaches \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The part \(B C D\) of the wire is smooth. The mass of the ring is 0.2 kg .
  2. (a) Find the speed of the ring at \(C\), where angle \(B O C = 30 ^ { \circ }\).
    (b) Find the greatest speed of the ring.