Questions — CAIE M1 (732 questions)

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CAIE M1 2015 November Q6
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-4_149_410_306_518} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-4_133_406_260_1210} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small ring of mass 0.024 kg is threaded on a fixed rough horizontal rod. A light inextensible string is attached to the ring and the string is pulled with a force of magnitude 0.195 N at an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac { 5 } { 13 }\). When the angle \(\theta\) is below the horizontal (see Fig. 1) the ring is in limiting equilibrium.
  1. Find the coefficient of friction between the ring and the rod. When the angle \(\theta\) is above the horizontal (see Fig. 2) the ring moves.
  2. Find the acceleration of the ring.
CAIE M1 2015 November Q7
7 A car of mass 1600 kg moves with constant power 14 kW as it travels along a straight horizontal road. The car takes 25 s to travel between two points \(A\) and \(B\) on the road.
  1. Find the work done by the car's engine while the car travels from \(A\) to \(B\). The resistance to the car's motion is constant and equal to 235 N . The car has accelerations at \(A\) and \(B\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. Find
  2. the gain in kinetic energy by the car in moving from \(A\) to \(B\),
  3. the distance \(A B\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2015 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_446_497_258_826} A small ball \(B\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 3 kg is attached to the other end of the string. The string passes over a fixed smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on a rough plane inclined to the horizontal at an angle of \(\alpha\), where \(\cos \alpha = 0.8\) (see diagram). State the tension in the string and find the normal component of the contact force exerted on \(B\) by the plane.
CAIE M1 2015 November Q2
5 marks
2
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_195_719_1114_712} A ring of mass 0.2 kg is threaded on a fixed rough horizontal rod and a light inextensible string is attached to the ring at an angle \(\alpha\) above the horizontal, where \(\cos \alpha = 0.96\). The ring is in limiting equilibrium with the tension in the string \(T \mathrm {~N}\) (see diagram). Given that the coefficient of friction between the ring and the rod is 0.25 , find the value of \(T\).
[0pt] [5]
CAIE M1 2015 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_296_735_1685_705} Three horizontal forces of magnitudes \(150 \mathrm {~N} , 100 \mathrm {~N}\) and \(P \mathrm {~N}\) have directions as shown in the diagram. The resultant of the three forces is shown by the broken line in the diagram. This resultant has magnitude 120 N and makes an angle \(75 ^ { \circ }\) with the 150 N force. Find the values of \(P\) and \(\theta\).
CAIE M1 2015 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-3_442_495_255_826} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is at rest with \(B\) held on the horizontal floor, the string taut and its straight parts vertical. \(A\) is at a height of 1.6 m above the floor (see diagram). \(B\) is released and the system begins to move; \(B\) does not reach the pulley. Find
  1. the acceleration of the particles and the tension in the string before \(A\) reaches the floor,
  2. the greatest height above the floor reached by \(B\).
CAIE M1 2015 November Q5
5 A cyclist and his bicycle have a total mass of 90 kg . The cyclist starts to move with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a straight hill, of length 500 m , which is inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal. The cyclist moves with constant acceleration until he reaches the bottom of the hill with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist generates 420 W of power while moving down the hill. The resistance to the motion of the cyclist and his bicycle, \(R \mathrm {~N}\), and the cyclist's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), both vary.
  1. Show that \(R = \frac { 420 } { v } + 43.56\).
  2. Find the cyclist's speed at the mid-point of the hill. Hence find the decrease in the value of \(R\) when the cyclist moves from the top of the hill to the mid-point of the hill, and when the cyclist moves from the mid-point of the hill to the bottom of the hill.
CAIE M1 2015 November Q6
6 A particle \(P\) starts from rest at a point \(O\) of a straight line and moves along the line. The displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(x \mathrm {~m}\), where $$x = 0.08 t ^ { 2 } - 0.0002 t ^ { 3 }$$
  1. Find the value of \(t\) when \(P\) returns to \(O\) and find the speed of \(P\) as it passes through \(O\) on its return.
  2. For the motion of \(P\) until the instant it returns to \(O\), find
    (a) the total distance travelled,
    (b) the average speed.
CAIE M1 2015 November Q7
7 A straight hill \(A B\) has length 400 m with \(A\) at the top and \(B\) at the bottom and is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. A straight horizontal road \(B C\) has length 750 m . A car of mass 1250 kg has a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) when starting to move down the hill. While moving down the hill the resistance to the motion of the car is 2000 N and the driving force is constant. The speed of the car on reaching \(B\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By using work and energy, find the driving force of the car. On reaching \(B\) the car moves along the road \(B C\). The driving force is constant and twice that when the car was on the hill. The resistance to the motion of the car continues to be 2000 N . Find
  2. the acceleration of the car while moving from \(B\) to \(C\),
  3. the power of the car's engine as the car reaches \(C\).
CAIE M1 2016 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-2_241_823_264_660} Two particles \(P\) and \(Q\), of masses 0.6 kg and 0.4 kg respectively, are connected by a light inextensible string. The string passes over a small smooth light pulley fixed at the edge of a smooth horizontal table. Initially \(P\) is held at rest on the table and \(Q\) hangs vertically (see diagram). \(P\) is then released. Find the tension in the string and the acceleration of \(Q\).
CAIE M1 2016 November Q2
2 A particle of mass 0.1 kg is released from rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal. It is given that, 5 seconds after release, the particle has a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
  1. Find the acceleration of the particle and hence show that the magnitude of the frictional force acting on the particle is 0.302 N , correct to 3 significant figures.
  2. Find the coefficient of friction between the particle and the plane.
CAIE M1 2016 November Q3
3 A particle \(P\) is projected vertically upwards from a point \(O\). When the particle is at a height of 0.5 m , its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the greatest height reached by the particle above \(O\),
  2. the time after projection at which the particle returns to \(O\).
CAIE M1 2016 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-2_334_832_1617_660} Three coplanar forces of magnitudes \(F \mathrm {~N} , 2 F \mathrm {~N}\) and 15 N act at a point \(P\), as shown in the diagram. Given that the forces are in equilibrium, find the values of \(F\) and \(\alpha\).
CAIE M1 2016 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_574_1205_260_470} The diagram shows a velocity-time graph which models the motion of a cyclist. The graph consists of five straight line segments. The cyclist accelerates from rest to a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 10 s , and then travels at this speed for a further 20 s . The cyclist then descends a hill, accelerating to speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 10 s . This speed is maintained for a further 30 s . The cyclist then decelerates to rest over a period of 20 s .
  1. Find the acceleration of the cyclist during the first 10 seconds.
  2. Show that the total distance travelled by the cyclist in the 90 seconds of motion may be expressed as \(( 45 V + 150 ) \mathrm { m }\). Hence find \(V\), given that the total distance travelled by the cyclist is 465 m .
  3. The combined mass of the cyclist and the bicycle is 80 kg . The cyclist experiences a constant resistance to motion of 20 N . Use an energy method to find the vertical distance which the cyclist descends during the downhill section from \(t = 30\) to \(t = 40\), assuming that the cyclist does no work during this time.
CAIE M1 2016 November Q6
4 marks
6 A block of mass 25 kg is pulled along horizontal ground by a force of magnitude 50 N inclined at \(10 ^ { \circ }\) above the horizontal. The block starts from rest and travels a distance of 20 m . There is a constant resistance force of magnitude 30 N opposing motion.
  1. Find the work done by the pulling force.
  2. Use an energy method to find the speed of the block when it has moved a distance of 20 m .
  3. Find the greatest power exerted by the 50 N force.
    \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_236_1027_2161_566} After the block has travelled the 20 m , it comes to a plane inclined at \(5 ^ { \circ }\) to the horizontal. The force of 50 N is now inclined at an angle of \(10 ^ { \circ }\) to the plane and pulls the block directly up the plane (see diagram). The resistance force remains 30 N .
  4. Find the time it takes for the block to come to rest from the instant when it reaches the foot of the inclined plane.
    [0pt] [4]
CAIE M1 2016 November Q7
7 A racing car is moving in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after the car starts from rest is given by $$\begin{array} { l l } a = 15 t - 3 t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 5
a = - \frac { 625 } { t ^ { 2 } } & \text { for } 5 < t \leqslant k \end{array}$$ where \(k\) is a constant.
  1. Find the maximum acceleration of the car in the first five seconds of its motion.
  2. Find the distance of the car from its starting point when \(t = 5\).
  3. The car comes to rest when \(t = k\). Find the value of \(k\).
CAIE M1 2016 November Q1
1 A particle of mass 2 kg is initially at rest on a rough horizontal plane. A force of magnitude 10 N is applied to the particle at \(15 ^ { \circ }\) above the horizontal. It is given that 10 s after the force is applied, the particle has a speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the magnitude of the frictional force is 8.96 N , correct to 3 significant figures.
  2. Find the coefficient of friction between the particle and the plane.
CAIE M1 2016 November Q2
2 A particle moves in a straight line. Its displacement \(t \mathrm {~s}\) after leaving a fixed point \(O\) on the line is \(s \mathrm {~m}\), where \(s = 2 t ^ { 2 } - \frac { 80 } { 3 } t ^ { \frac { 3 } { 2 } }\).
  1. Find the time at which the acceleration of the particle is zero.
  2. Find the displacement and velocity of the particle at this instant.
CAIE M1 2016 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{221f995d-2ef2-4be8-935b-49f8acf1cbe0-2_317_1104_1050_518} A boat is being pulled along a river by two people. One of the people walks along a path on one side of the river and the other person walks along a path on the opposite side of the river. The first person exerts a horizontal force of 60 N at an angle of \(25 ^ { \circ }\) to the direction of the river. The second person exerts a horizontal force of 50 N at an angle of \(15 ^ { \circ }\) to the direction of the river (see diagram).
  1. Find the total force exerted by the two people in the direction of the river.
  2. Find the magnitude and direction of the resultant force exerted by the two people.
CAIE M1 2016 November Q4
4 A girl on a sledge starts, with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at the top of a slope of length 100 m which is at an angle of \(20 ^ { \circ }\) to the horizontal. The sledge slides directly down the slope.
  1. Given that there is no resistance to the sledge's motion, find the speed of the sledge at the bottom of the slope.
  2. It is given instead that the sledge experiences a resistance to motion such that the total work done against the resistance is 8500 J , and the speed of the sledge at the bottom of the slope is \(21 \mathrm {~ms} ^ { - 1 }\). Find the total mass of the girl and the sledge.
CAIE M1 2016 November Q5
5 A particle of mass \(m \mathrm {~kg}\) is resting on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. A force of magnitude 10 N applied to the particle up a line of greatest slope of the plane is just sufficient to stop the particle sliding down the plane. When a force of 75 N is applied to the particle up a line of greatest slope of the plane, the particle is on the point of sliding up the plane. Find \(m\) and the coefficient of friction between the particle and the plane.
CAIE M1 2016 November Q6
6 A van of mass 3000 kg is pulling a trailer of mass 500 kg along a straight horizontal road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The system of the van and the trailer is modelled as two particles connected by a light inextensible cable. There is a constant resistance to motion of 300 N on the van and 100 N on the trailer.
  1. Find the power of the van's engine.
  2. Write down the tension in the cable. The van reaches the bottom of a hill inclined at \(4 ^ { \circ }\) to the horizontal with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the van's engine is increased to 25000 W .
  3. Assuming that the resistance forces remain the same, find the new tension in the cable at the instant when the speed of the van up the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2016 November Q7
7 A car starts from rest and moves in a straight line from point \(A\) with constant acceleration \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 10 s . The car then travels at constant speed for 30 s before decelerating uniformly, coming to rest at point \(B\). The distance \(A B\) is 1.5 km .
  1. Find the total distance travelled in the first 40 s of motion. When the car has been moving for 20 s , a motorcycle starts from rest and accelerates uniformly in a straight line from point \(A\) to a speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then maintains this speed for 30 s before decelerating uniformly to rest at point \(B\). The motorcycle comes to rest at the same time as the car.
  2. Given that the magnitude of the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the motorcycle is three times the magnitude of its deceleration, find the value of \(a\).
  3. Sketch the displacement-time graph for the motion of the car.
CAIE M1 2016 November Q1
1 A crane is used to raise a block of mass 50 kg vertically upwards at constant speed through a height of 3.5 m . There is a constant resistance to motion of 25 N .
  1. Find the work done by the crane.
  2. Given that the time taken to raise the block is 2 s , find the power of the crane.
CAIE M1 2016 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_342_629_616_756} The diagram shows a small object \(P\) of mass 20 kg held in equilibrium by light ropes attached to fixed points \(A\) and \(B\). The rope \(P A\) is inclined at an angle of \(50 ^ { \circ }\) above the horizontal, the rope \(P B\) is inclined at an angle of \(10 ^ { \circ }\) below the horizontal, and both ropes are in the same vertical plane. Find the tension in the rope \(P A\) and the tension in the rope \(P B\).