Questions M1 (1912 questions)

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CAIE M1 2019 November Q2
5 marks Standard +0.3
2 A train of mass 150000 kg ascends a straight slope inclined at \(\alpha ^ { \circ }\) to the horizontal with a constant driving force of 16000 N . At a point \(A\) on the slope the speed of the train is \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Point \(B\) on the slope is 500 m beyond \(A\). At \(B\) the speed of the train is \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a resistance force acting on the train and the train does \(4 \times 10 ^ { 6 } \mathrm {~J}\) of work against this resistance force between \(A\) and \(B\). Find the value of \(\alpha\).
CAIE M1 2019 November Q3
6 marks Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-05_479_647_264_749} Three coplanar forces of magnitudes \(50 \mathrm {~N} , 60 \mathrm {~N}\) and 100 N act at a point. The resultant of the forces has magnitude \(R \mathrm {~N}\). The directions of these forces are shown in the diagram. Find the values of \(R\) and \(\alpha\).
CAIE M1 2019 November Q4
6 marks Standard +0.3
4 A car travels along a straight road with constant acceleration. It passes through points \(P , Q , R\) and \(S\). The times taken for the car to travel from \(P\) to \(Q , Q\) to \(R\) and \(R\) to \(S\) are each equal to 10 s . The distance \(Q R\) is 1.5 times the distance \(P Q\). At point \(Q\) the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the acceleration of the car is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the distance \(Q S\) and hence find the average speed of the car between \(Q\) and \(S\).
CAIE M1 2019 November Q5
8 marks Moderate -0.3
5 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and his bicycle is 80 kg . His power output is a constant 240 W . His acceleration when he is travelling at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that the resistance to the cyclist's motion is 16 N .
  2. Find the steady speed that the cyclist can maintain if his power output and the resistance force are both unchanged.
  3. The cyclist later ascends a straight hill inclined at \(3 ^ { \circ }\) to the horizontal. His power output and the resistance force are still both unchanged. Find his acceleration when he is travelling at \(4 \mathrm {~ms} ^ { - 1 }\).
CAIE M1 2019 November Q6
9 marks Standard +0.3
6 Particle \(P\) travels in a straight line from \(A\) to \(B\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.04 t ^ { 3 } + c t ^ { 2 } + k t$$ \(P\) takes 5 s to travel from \(A\) to \(B\) and it reaches \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is 25 m .
  1. Find the values of the constants \(c\) and \(k\).
  2. Show that the acceleration of \(P\) is a minimum when \(t = 2.5\).
CAIE M1 2019 November Q7
13 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-12_565_511_260_817} Two particles \(A\) and \(B\) have masses \(m \mathrm {~kg}\) and \(k m \mathrm {~kg}\) respectively, where \(k > 1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.81 m above horizontal ground (see diagram). The system is released from rest and particle \(B\) reaches the ground 0.9 s later. The particle \(A\) does not reach the pulley in its subsequent motion.
  1. Find the value of \(k\) and show that the tension in the string before \(B\) reaches the ground is equal to \(12 m \mathrm {~N}\).
    At the instant when \(B\) reaches the ground, the string breaks.
  2. Show that the speed of \(A\) when it reaches the ground is \(5.97 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures, and find the time taken, after the string breaks, for \(A\) to reach the ground.
  3. Sketch a velocity-time graph for the motion of particle \(A\) from the instant when the system is released until \(A\) reaches the ground. 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 Specimen Q1
4 marks Easy -1.2
1 A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.
  1. Find the work done by the weightlifter.
  2. Given that the time taken to raise the mass is 1.2 s , find the average power developed by the weightlifter.
CAIE M1 Specimen Q2
6 marks Moderate -0.3
2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
  1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
  2. Find the distance that the particle travels along the ground before it comes to rest.
CAIE M1 Specimen Q3
6 marks Moderate -0.3
3 A lorry of mass 24000 kg is travelling up a hill which is inclined at \(3 ^ { \circ }\) to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N .
  1. When the speed of the lorry is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power developed by the lorry's engine.
  2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N .
    \includegraphics[max width=\textwidth, alt={}, center]{75c345bb-7cbd-4b2a-b3a0-0086b80b36c1-05_499_784_258_685} Blocks \(P\) and \(Q\), of mass \(m \mathrm {~kg}\) and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2 . Find the set of values of \(m\) for which the two blocks remain at rest.
    \includegraphics[max width=\textwidth, alt={}, center]{75c345bb-7cbd-4b2a-b3a0-0086b80b36c1-06_351_1038_255_557} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(A B\) of length 3 m . Three horizontal forces of magnitudes \(F \mathrm {~N} , 10 \mathrm {~N}\) and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R \mathrm {~N}\) in the direction shown in the diagram.
CAIE M1 Specimen Q6
10 marks Moderate -0.3
6 A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) after leaving \(O\) is given by $$v = 0.6 t - 0.03 t ^ { 2 }$$
  1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time.
  2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity.
CAIE M1 Specimen Q7
10 marks Standard +0.3
7 A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 36 m . The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(A B\) is 210 m .
  1. Find the total time that the cyclist takes to travel from \(A\) to \(B\).
    24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
  2. Find the time that it takes from when the cyclist starts until the car overtakes her.
Edexcel M1 2010 January Q1
6 marks Moderate -0.8
  1. A particle \(A\) of mass 2 kg is moving along a straight horizontal line with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(B\) of mass \(m \mathrm {~kg}\) is moving along the same straight line, in the opposite direction to \(A\), with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particles collide. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, \(A\) is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
    1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
    2. the value of \(m\).
    3. An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s .
    4. In the space below, sketch a speed-time graph to illustrate the motion of the athlete.
    5. Calculate the value of \(T\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{330c2068-fe0a-4c6d-b892-79ab173c6a11-04_271_750_214_598} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A particle of mass \(m \mathrm {~kg}\) is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively, as shown in Figure 1. Given that the tension in \(A C\) is 20 N , find
  2. the tension in \(B C\),
  3. the value of \(m\).
Edexcel M1 2010 January Q4
10 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{330c2068-fe0a-4c6d-b892-79ab173c6a11-05_557_673_127_646} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A pole \(A B\) has length 3 m and weight \(W\) newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points \(A\) and \(C\) where \(A C = 1.8 \mathrm {~m}\), as shown in Figure 2. A load of weight 20 N is attached to the rod at \(B\). The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle.
  1. Show that the tension in the rope attached to the pole at \(C\) is \(\left( \frac { 5 } { 6 } W + \frac { 100 } { 3 } \right) \mathrm { N }\).
  2. Find, in terms of \(W\), the tension in the rope attached to the pole at \(A\). Given that the tension in the rope attached to the pole at \(C\) is eight times the tension in the rope attached to the pole at \(A\),
  3. find the value of \(W\).
Edexcel M1 2010 January Q5
15 marks Standard +0.3
  1. A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find
    1. the acceleration of the particle,
    2. the coefficient of friction between the particle and the plane.
    The particle is now held on the same rough plane by a horizontal force of magnitude \(X\) newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{330c2068-fe0a-4c6d-b892-79ab173c6a11-07_255_725_890_621} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure}
  2. Find the value of \(X\).
Edexcel M1 2010 January Q6
14 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{330c2068-fe0a-4c6d-b892-79ab173c6a11-09_519_537_210_708} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two particles \(A\) and \(B\) have masses \(5 m\) and \(k m\) respectively, where \(k < 5\). The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut, the hanging parts of the string vertical and with \(A\) and \(B\) at the same height above a horizontal plane, as shown in Figure 4. The system is released from rest. After release, \(A\) descends with acceleration \(\frac { 1 } { 4 } g\).
  1. Show that the tension in the string as \(A\) descends is \(\frac { 15 } { 4 } \mathrm { mg }\).
  2. Find the value of \(k\).
  3. State how you have used the information that the pulley is smooth. After descending for 1.2 s , the particle \(A\) reaches the plane. It is immediately brought to rest by the impact with the plane. The initial distance between \(B\) and the pulley is such that, in the subsequent motion, \(B\) does not reach the pulley.
  4. Find the greatest height reached by \(B\) above the plane.
Edexcel M1 2010 January Q7
14 marks Moderate -0.3
7. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving along a straight line with constant velocity. At time \(t\) hours the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 9 \mathbf { i } - 6 \mathbf { j }\). When \(t = 4 , \mathbf { s } = 21 \mathbf { i } + 10 \mathbf { j }\). Find
  1. the speed of \(S\),
  2. the direction in which \(S\) is moving, giving your answer as a bearing.
  3. Show that \(\mathbf { s } = ( 3 t + 9 ) \mathbf { i } + ( 4 t - 6 ) \mathbf { j }\). A lighthouse \(L\) is located at the point with position vector \(( 18 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km }\). When \(t = T\), the ship \(S\) is 10 km from \(L\).
  4. Find the possible values of \(T\).
Edexcel M1 2011 January Q1
5 marks Moderate -0.8
  1. Two particles \(B\) and \(C\) have mass \(m \mathrm {~kg}\) and 3 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table. The two particles collide directly. Immediately before the collision, the speed of \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the direction of motion of \(C\) is reversed and the direction of motion of \(B\) is unchanged. Immediately after the collision, the speed of \(B\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find
  1. the value of \(m\),
  2. the magnitude of the impulse received by \(C\).
Edexcel M1 2011 January Q2
8 marks Moderate -0.3
2. A ball is thrown vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(P\) at height \(h\) metres above the ground. The ball hits the ground 0.75 s later. The speed of the ball immediately before it hits the ground is \(6.45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is modelled as a particle.
  1. Show that \(u = 0.9\)
  2. Find the height above \(P\) to which the ball rises before it starts to fall towards the ground again.
  3. Find the value of \(h\).
Edexcel M1 2011 January Q3
10 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-04_245_860_260_543} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform beam \(A B\) has mass 20 kg and length 6 m . The beam rests in equilibrium in a horizontal position on two smooth supports. One support is at \(C\), where \(A C = 1 \mathrm {~m}\), and the other is at the end \(B\), as shown in Figure 1. The beam is modelled as a rod.
  1. Find the magnitudes of the reactions on the beam at \(B\) and at \(C\). A boy of mass 30 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The magnitudes of the reactions on the beam at \(B\) and at \(C\) are now equal. The boy is modelled as a particle.
  2. Find the distance \(A D\).
Edexcel M1 2011 January Q4
11 marks Moderate -0.5
  1. A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. The velocity of \(P\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 0\), and \(( 7 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 5 \mathrm {~s}\).
Find
  1. the speed of \(P\) at \(t = 0\),
  2. the vector \(\mathbf { F }\) in the form \(a \mathbf { i } + b \mathbf { j }\),
  3. the value of \(t\) when \(P\) is moving parallel to \(\mathbf { i }\).
Edexcel M1 2011 January Q5
10 marks Moderate -0.8
  1. A car accelerates uniformly from rest for 20 seconds. It moves at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 40 seconds and then decelerates uniformly for 10 seconds until it comes to rest.
    1. For the motion of the car, sketch
      1. a speed-time graph,
      2. an acceleration-time graph.
    Given that the total distance moved by the car is 880 m ,
  2. find the value of \(v\).
Edexcel M1 2011 January Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-10_426_768_239_653} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of weight 120 N is placed on a fixed rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
The particle is held at rest in equilibrium by a horizontal force of magnitude 30 N , which acts in the vertical plane containing the line of greatest slope of the plane through the particle, as shown in Figure 2.
  1. Show that the normal reaction between the particle and the plane has magnitude 114 N . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-10_433_774_1464_604} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The horizontal force is removed and replaced by a force of magnitude \(P\) newtons acting up the slope along the line of greatest slope of the plane through the particle, as shown in Figure 3. The particle remains in equilibrium.
  2. Find the greatest possible value of \(P\).
  3. Find the magnitude and direction of the frictional force acting on the particle when \(P = 30\).
Edexcel M1 2011 January Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-12_581_1211_235_370} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two particles \(A\) and \(B\), of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially \(B\) is held at rest on a rough fixed plane inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The part of the string from \(B\) to \(P\) is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, \(P\), fixed at the top of the plane. The particle \(A\) hangs freely below \(P\), as shown in Figure 4. The coefficient of friction between \(B\) and the plane is \(\frac { 2 } { 3 }\). The particles are released from rest with the string taut and \(B\) moves up the plane.
  1. Find the magnitude of the acceleration of \(B\) immediately after release.
  2. Find the speed of \(B\) when it has moved 1 m up the plane. When \(B\) has moved 1 m up the plane the string breaks. Given that in the subsequent motion \(B\) does not reach \(P\),
  3. find the time between the instants when the string breaks and when \(B\) comes to instantaneous rest.
Edexcel M1 2012 January Q1
5 marks Moderate -0.8
  1. A railway truck \(P\), of mass \(m \mathrm {~kg}\), is moving along a straight horizontal track with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Truck \(P\) collides with a truck \(Q\) of mass 3000 kg , which is at rest on the same track. Immediately after the collision the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is reversed by the collision.
Modelling the trucks as particles, find
  1. the magnitude of the impulse exerted by \(P\) on \(Q\),
  2. the value of \(m\).
Edexcel M1 2012 January Q2
6 marks Moderate -0.8
2. A car of mass 1000 kg is towing a caravan of mass 750 kg along a straight horizontal road. The caravan is connected to the car by a tow-bar which is parallel to the direction of motion of the car and the caravan. The tow-bar is modelled as a light rod. The engine of the car provides a constant driving force of 3200 N . The resistances to the motion of the car and the caravan are modelled as constant forces of magnitude 800 newtons and \(R\) newtons respectively. Given that the acceleration of the car and the caravan is \(0.88 \mathrm {~ms} ^ { - 2 }\),
  1. show that \(R = 860\),
  2. find the tension in the tow-bar.