Questions — CAIE M1 (732 questions)

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CAIE M1 2016 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_312_1207_1320_468} Particles \(P\) and \(Q\), of masses 7 kg and 3 kg respectively, are attached to the two ends of a light inextensible string. The string passes over two small smooth pulleys attached to the two ends of a horizontal table. The two particles hang vertically below the two pulleys. The two particles are both initially at rest, 0.5 m below the level of the table, and 0.4 m above the horizontal floor (see diagram).
  1. Find the acceleration of the particles and the speed of \(P\) immediately before it reaches the floor.
  2. Determine whether \(Q\) comes to instantaneous rest before it reaches the pulley directly above it.
CAIE M1 2016 November Q4
4 A ball \(A\) is released from rest at the top of a tall tower. One second later, another ball \(B\) is projected vertically upwards from ground level near the bottom of the tower with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The two balls are at the same height 1.5 s after ball \(B\) is projected.
  1. Show that the height of the tower is 50 m .
  2. Find the length of time for which ball \(B\) has been in motion when ball \(A\) reaches the ground. Hence find the total distance travelled by ball \(B\) up to the instant when ball \(A\) reaches the ground.
CAIE M1 2016 November Q5
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
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
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
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
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
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
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
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
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
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 Q1
1 marks
1 A particle of mass 0.2 kg is resting in equilibrium on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal.
[0pt]
  1. Show that the friction force acting on the particle is 0.684 N , correct to 3 significant figures. [1]
    The coefficient of friction between the particle and the plane is 0.6 . A force of magnitude 0.9 N is applied to the particle down a line of greatest slope of the plane. The particle accelerates down the plane.
  2. Find this acceleration.
    \includegraphics[max width=\textwidth, alt={}, center]{d3bb18c3-8c6f-41d4-808e-479f0c92250f-03_250_657_258_744} A block of mass 15 kg hangs in equilibrium below a horizontal ceiling attached to two strings as shown in the diagram. One of the strings is inclined at \(45 ^ { \circ }\) to the horizontal and the tension in this string is 120 N . The other string is inclined at \(\theta ^ { \circ }\) to the horizontal and the tension in this string is \(T \mathrm {~N}\). Find the values of \(T\) and \(\theta\).
CAIE M1 2017 November Q3
4 marks
3 A car travels along a straight road with constant acceleration. It passes through points \(A , B\) and \(C\). The car passes point \(A\) with velocity \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The two sections \(A B\) and \(B C\) are of equal length. The times taken to travel along \(A B\) and \(B C\) are 5 s and 3 s respectively.
  1. Write down an expression for the distance \(A B\) in terms of the acceleration of the car. Write down a similar expression for the distance \(A C\). Hence show that the acceleration of the car is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    [0pt] [4]
    \includegraphics[max width=\textwidth, alt={}, center]{d3bb18c3-8c6f-41d4-808e-479f0c92250f-04_67_1575_580_324}
    ...................................................................................................................................
  2. Find the speed of the car as it passes point \(C\).
CAIE M1 2017 November Q4
4 A particle \(P\) is projected vertically upwards from horizontal ground with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the time taken for \(P\) to return to the ground.
    The time in seconds after \(P\) is projected is denoted by \(t\). When \(t = 1\), a second particle \(Q\) is projected vertically upwards with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point which is 5 m above the ground. Particles \(P\) and \(Q\) move in different vertical lines.
  2. Find the set of values of \(t\) for which the two particles are moving in the same direction.
CAIE M1 2017 November Q5
5 A cyclist is riding up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.04\). The total mass of the bicycle and rider is 80 kg . The cyclist is riding at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force resisting the motion. The work done by the cyclist against this resistance force over a distance of 25 m is 600 J .
  1. Find the power output of the cyclist.
    The cyclist reaches the top of the hill, where the road becomes horizontal, with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist continues to work at the same rate on the horizontal part of the road.
  2. Find the speed of the cyclist 10 seconds after reaching the top of the hill, given that the work done by the cyclist during this period against the resistance force is 1200 J .
    \includegraphics[max width=\textwidth, alt={}, center]{d3bb18c3-8c6f-41d4-808e-479f0c92250f-08_346_616_260_762} Two particles \(P\) and \(Q\), each of mass \(m \mathrm {~kg}\), 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 rough plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). Particle \(P\) rests on the plane and particle \(Q\) hangs vertically, as shown in the diagram. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. The system is in limiting equilibrium.
CAIE M1 2017 November Q7
7 A particle starts from rest and moves in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after the start is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = - 0.01 t ^ { 3 } + 0.22 t ^ { 2 } - 0.4 t$$
  1. Find the two positive values of \(t\) for which the particle is instantaneously at rest.
  2. Find the time at which the acceleration of the particle is greatest.
  3. Find the distance travelled by the particle while its velocity is positive.
CAIE M1 2017 November Q2
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
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
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
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
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).
  5. 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 \(A\).
  6. Use an energy method to find the speed of \(P\) at \(A\).
CAIE M1 2018 November Q1
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
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
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.