Questions — CAIE (7646 questions)

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CAIE M1 2016 November Q5
8 marks Moderate -0.3
5 A particle \(P\) starts from a fixed point \(O\) and moves in a straight line. At time \(t\) s after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by \(v = 6 t - 0.3 t ^ { 2 }\). The particle comes to instantaneous rest at point \(X\).
  1. Find the distance \(O X\). A second particle \(Q\) starts from rest from \(O\), at the same instant as \(P\), and also travels in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(Q\) is given by \(a = k - 12 t\), where \(k\) is a constant. The displacement of \(Q\) from \(O\) is 400 m when \(t = 10\).
  2. Find the value of \(k\).
CAIE M1 2016 November Q6
9 marks Standard +0.3
6 A cyclist is cycling with constant power of 160 W along a horizontal straight road. There is a constant resistance to motion of 20 N . At an instant when the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that the total mass of the cyclist and bicycle is 80 kg . The cyclist comes to a hill inclined at \(2 ^ { \circ }\) to the horizontal. When the cyclist starts climbing the hill, he increases his power to a constant 300 W . The resistance to motion remains 20 N .
  2. Show that the steady speed up the hill which the cyclist can maintain when working at this power is \(6.26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  3. Find the acceleration at an instant when the cyclist is travelling at \(90 \%\) of the speed in part (ii).
CAIE M1 2016 November Q7
9 marks Standard +0.3
7 A box of mass 50 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal.
  1. Find an inequality for the coefficient of friction between the box and the plane. In fact the coefficient of friction between the box and the plane is 0.19 .
  2. A girl pushes the box with a force of 50 N , acting down a line of greatest slope of the plane, for a distance of 5 m . She then stops pushing. Use an energy method to find the speed of the box when it has travelled a further 5 m . The box then comes to a plane inclined at \(20 ^ { \circ }\) below the horizontal. The box moves down a line of greatest slope of this plane. The coefficient of friction is still 0.19 and the girl is not pushing the box.
  3. Find the acceleration of the box.
CAIE M1 2017 November Q1
4 marks Moderate -0.3
1 A block of mass 3 kg is initially at rest on a smooth horizontal floor. A force of 12 N , acting at an angle of \(25 ^ { \circ }\) above the horizontal, is applied to the block. Find the distance travelled by the block in the first 5 seconds of its motion.
CAIE M1 2017 November Q2
6 marks Moderate -0.3
2 A tractor of mass 3700 kg is travelling along a straight horizontal road at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total resistance to motion is 1150 N .
  1. Find the power output of the tractor's engine.
    The tractor comes to a hill inclined at \(4 ^ { \circ }\) above the horizontal. The power output is increased to 25 kW and the resistance to motion is unchanged.
  2. Find the deceleration of the tractor at the instant it begins to climb the hill.
  3. Find the constant speed that the tractor could maintain on the hill when working at this power.
CAIE M1 2017 November Q3
6 marks Moderate -0.3
3 A roller-coaster car (including passengers) has a mass of 840 kg . The roller-coaster ride includes a section where the car climbs a straight ramp of length 8 m inclined at \(30 ^ { \circ }\) above the horizontal. The car then immediately descends another ramp of length 10 m inclined at \(20 ^ { \circ }\) below the horizontal. The resistance to motion acting on the car is 640 N throughout the motion.
  1. Find the total work done against the resistance force as the car ascends the first ramp and descends the second ramp.
  2. The speed of the car at the bottom of the first ramp is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Use an energy method to find the speed of the car when it reaches the bottom of the second ramp.
CAIE M1 2017 November Q4
7 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{db1b5f31-1a41-44dd-ae9a-0c67336997eb-05_600_1155_262_497} The diagram shows the velocity-time graph of a particle which moves in a straight line. The graph consists of 5 straight line segments. The particle starts from rest at a point \(A\) at time \(t = 0\), and initially travels towards point \(B\) on the line.
  1. Show that the acceleration of the particle between \(t = 3.5\) and \(t = 6\) is \(- 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. The acceleration of the particle between \(t = 6\) and \(t = 10\) is \(7.5 \mathrm {~ms} ^ { - 2 }\). When \(t = 10\) the velocity of the particle is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(V\).
  3. The particle comes to rest at \(B\) at time \(T\) s. Given that the total distance travelled by the particle between \(t = 0\) and \(t = T\) is 100 m , find the value of \(T\).
CAIE M1 2017 November Q5
8 marks Standard +0.3
5 A particle starts from a point \(O\) and moves in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$\begin{array} { l l } v = 1.5 + 0.4 t & \text { for } 0 \leqslant t \leqslant 5 , \\ v = \frac { 100 } { t ^ { 2 } } - 0.1 t & \text { for } t \geqslant 5 . \end{array}$$
  1. Find the acceleration of the particle during the first 5 seconds of motion.
  2. Find the value of \(t\) when the particle is instantaneously at rest.
  3. Find the total distance travelled by the particle in the first 10 seconds of motion.
CAIE M1 2017 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{db1b5f31-1a41-44dd-ae9a-0c67336997eb-08_529_606_260_767} Coplanar forces, of magnitudes \(F \mathrm {~N} , 3 F \mathrm {~N} , G \mathrm {~N}\) and 50 N , act at a point \(P\), as shown in the diagram.
  1. Given that \(F = 0 , G = 75\) and \(\alpha = 60 ^ { \circ }\), find the magnitude and direction of the resultant force.
  2. Given instead that \(G = 0\) and the forces are in equilibrium, find the values of \(F\) and \(\alpha\).
CAIE M1 2017 November Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{db1b5f31-1a41-44dd-ae9a-0c67336997eb-10_212_1029_255_557} Two particles \(A\) and \(B\) of masses 0.9 kg and 0.4 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 top of two inclined planes. The particles are initially at rest with \(A\) on a smooth plane inclined at angle \(\theta ^ { \circ }\) to the horizontal and \(B\) on a plane inclined at angle \(25 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 2.5 N is applied to \(B\) acting down the plane (see diagram).
  1. For the case where \(\theta = 15\) and the plane on which \(B\) rests is smooth, find the acceleration of \(B\).
  2. For a different value of \(\theta\), the plane on which \(B\) rests is rough with coefficient of friction between the plane and \(B\) of 0.8 . The system is in limiting equilibrium with \(B\) on the point of moving in the direction of the 2.5 N force. Find the value of \(\theta\).
CAIE M1 2017 November Q2
6 marks Moderate -0.8
2 A lorry of mass 7850 kg travels on a straight hill which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of 1480 N .
  1. Find the power of the lorry's engine when the lorry is going up the hill at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the power of the lorry's engine at an instant when the lorry is going down the hill at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with an acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2017 November Q3
6 marks Moderate -0.8
3 A particle is released from rest and slides down a line of greatest slope of a rough plane which is inclined at \(25 ^ { \circ }\) 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 travelled by the particle in the first 3 s after it is released.
CAIE M1 2017 November Q4
7 marks Standard +0.3
4 Two particles \(A\) and \(B\) have masses 0.35 kg and 0.45 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small fixed smooth pulley which is 1 m above horizontal ground. Initially particle \(A\) is held at rest on the ground vertically below the pulley, with the string taut. Particle \(B\) hangs vertically below the pulley at a height of 0.64 m above the ground. Particle \(A\) is released.
  1. Find the speed of \(A\) at the instant that \(B\) reaches the ground.
  2. Assuming that \(B\) does not bounce after it reaches the ground, find the total distance travelled by \(A\) between the instant that \(B\) reaches the ground and the instant when the string becomes taut again.
CAIE M1 2017 November Q5
8 marks Moderate -0.3
5 A particle starts from a fixed origin with velocity \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle \(t \mathrm {~s}\) after it leaves the origin is given by \(a = k \left( 3 t ^ { 2 } - 12 t + 2 \right)\), where \(k\) is a constant. When \(t = 1\), the velocity of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the value of \(k\) is 0.1 .
  2. Find an expression for the displacement of the particle from the origin in terms of \(t\).
  3. Hence verify that the particle is again at the origin at \(t = 2\).
CAIE M1 2017 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{f08a4870-9466-4f8b-bd0f-431fb1803514-08_661_1244_262_452} The diagram shows the velocity-time graphs for two particles, \(P\) and \(Q\), which are moving in the same straight line. The graph for \(P\) consists of four straight line segments. The graph for \(Q\) consists of three straight line segments. Both particles start from the same initial position \(O\) on the line. \(Q\) starts 2 seconds after \(P\) and both particles come to rest at time \(t = T\). The greatest velocity of \(Q\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the displacement of \(P\) from \(O\) at \(t = 10\).
  2. Find the velocity of \(P\) at \(t = 12\).
  3. Given that the total distance covered by \(P\) during the \(T\) seconds of its motion is 49.5 m , find the value of \(T\).
  4. Given also that the acceleration of \(Q\) from \(t = 2\) to \(t = 6\) is \(1.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the value of \(V\) and hence find the distance between the two particles when they both come to rest at \(t = T\). \includegraphics[max width=\textwidth, alt={}, center]{f08a4870-9466-4f8b-bd0f-431fb1803514-10_392_529_262_808} A particle \(P\) of mass 0.2 kg rests on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.3 . A force of magnitude \(T \mathrm {~N}\) acts upwards on \(P\) at \(15 ^ { \circ }\) above a line of greatest slope of the plane (see diagram).
  1. Find the least value of \(T\) for which the particle remains at rest.
    The force of magnitude \(T \mathrm {~N}\) is now removed. A new force of magnitude 0.25 N acts on \(P\) up the plane, parallel to a line of greatest slope of the plane. Starting from rest, \(P\) slides down the plane. After moving a distance of \(3 \mathrm {~m} , P\) passes through the point \(A\).
  2. Use an energy method to find the speed of \(P\) at \(A\).
CAIE M1 2018 November Q1
4 marks Moderate -0.3
1 A particle of mass 0.2 kg moving in a straight line experiences a constant resistance force of 1.5 N . When the particle is moving at speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), a constant force of magnitude \(F \mathrm {~N}\) is applied to it in the direction in which it is moving. Given that the speed of the particle 5 seconds later is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(F\).
CAIE M1 2018 November Q2
4 marks Moderate -0.8
2 A high-speed train of mass 490000 kg is moving along a straight horizontal track at a constant speed of \(85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engines are supplying 4080 kW of power.
  1. Show that the resistance force is 48000 N .
  2. The train comes to a hill inclined at an angle \(\theta ^ { \circ }\) above the horizontal, where \(\sin \theta ^ { \circ } = \frac { 1 } { 200 }\). Given that the resistance force is unchanged, find the power required for the train to keep moving at the same constant speed of \(85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2018 November Q3
5 marks Moderate -0.3
3 A van of mass 2500 kg descends a hill of length 0.4 km inclined at \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of 600 N and the speed of the van increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it descends the hill. Find the work done by the van's engine as it descends the hill.
CAIE M1 2018 November Q4
6 marks Standard +0.3
4 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang freely below it. The system is released from rest, with both particles 0.8 m above horizontal ground. Particle \(A\) reaches the ground with a speed of \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the tension in the string during the motion before \(A\) reaches the ground.
  2. Find the value of \(m\).
CAIE M1 2018 November Q5
9 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{98a5537b-d503-4a42-bbfe-0bd221084ee0-06_449_654_260_742} Coplanar forces, of magnitudes \(15 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , act at a point \(B\) on the line \(A B C\) in the directions shown in the diagram.
  1. Find the magnitude and direction of the resultant force.
  2. The force of magnitude 15 N is now replaced by a force of magnitude \(F \mathrm {~N}\) acting in the same direction. The new resultant force has zero component in the direction \(B C\). Find the value of \(F\), and find also the magnitude and direction of the new resultant force.
CAIE M1 2018 November Q6
10 marks Standard +0.3
6 A particle is projected from a point \(P\) with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope \(P Q R\) of a rough inclined plane. The distances \(P Q\) and \(Q R\) are both equal to 0.8 m . The particle takes 0.6 s to travel from \(P\) to \(Q\) and 1 s to travel from \(Q\) to \(R\).
  1. Show that the deceleration of the particle is \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and hence find \(u\), giving your answer as an exact fraction.
  2. Given that the plane is inclined at \(3 ^ { \circ }\) to the horizontal, find the value of the coefficient of friction between the particle and the plane.
CAIE M1 2018 November Q7
12 marks Standard +0.3
7 A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 5.4 - 1.62 t$$
  1. Find the positive value of \(t\) at which the velocity of the particle is zero, giving your answer as an exact fraction.
  2. Find the velocity of the particle at \(t = 10\) and sketch the velocity-time graph for the first ten seconds of the motion.
  3. Find the total distance travelled during the first ten seconds of the motion.
    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 November Q3
5 marks Moderate -0.3
3 A particle of mass 1.2 kg moves in a straight line \(A B\). It is projected with speed \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) towards \(B\) and experiences a resistance force. The work done against this resistance force in moving from \(A\) to \(B\) is 25 J .
  1. Given that \(A B\) is horizontal, find the speed of the particle at \(B\).
  2. It is given instead that \(A B\) is inclined at \(30 ^ { \circ }\) below the horizontal and that the speed of the particle at \(B\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against the resistance force remains the same. Find the distance \(A B\).
CAIE M1 2018 November Q4
7 marks Moderate -0.3
4 A runner sets off from a point \(P\) at time \(t = 0\), where \(t\) is in seconds. The runner starts from rest and accelerates at \(1.2 \mathrm {~ms} ^ { - 2 }\) for 5 s . For the next 12 s the runner moves at constant speed before decelerating uniformly over a period of 3 s , coming to rest at \(Q\). A cyclist sets off from \(P\) at time \(t = 10\) and accelerates uniformly for 10 s , before immediately decelerating uniformly to rest at \(Q\) at time \(t = 30\).
  1. Sketch the velocity-time graph for the runner and show that the distance \(P Q\) is 96 m . \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-06_821_1451_708_388}
  2. Find the magnitude of the acceleration of the cyclist.
CAIE M1 2018 November Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-08_538_414_260_868} Two particles \(P\) and \(Q\), 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 with the particles hanging freely below it. \(Q\) is held at rest with the string taut at a height of \(h \mathrm {~m}\) above a horizontal floor (see diagram). \(Q\) is now released and both particles start to move. The pulley is sufficiently high so that \(P\) does not reach it at any stage. The time taken for \(Q\) to reach the floor is 0.6 s .
  1. Find the acceleration of \(Q\) before it reaches the floor and hence find the value of \(h\). \(Q\) remains at rest when it reaches the floor, and \(P\) continues to move upwards.
  2. Find the velocity of \(P\) at the instant when \(Q\) reaches the floor and the total time taken from the instant at which \(Q\) is released until the string becomes taut again.