Questions M1 (2067 questions)

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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.
CAIE M1 2018 November Q6
10 marks Moderate -0.3
6 A van of mass 3200 kg travels along a horizontal road. The power of the van's engine is constant and equal to 36 kW , and there is a constant resistance to motion acting on the van.
  1. When the speed of the van is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the resistance force.
    When the van is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it begins to ascend a hill inclined at \(1.5 ^ { \circ }\) to the horizontal. The power is increased and the resistance force is still equal to the value found in part (i).
  2. Find the power required to maintain this speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The engine is now stopped, with the van still travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the van decelerates to rest. Find the distance the van moves up the hill from the point at which the engine is stopped until it comes to rest.
CAIE M1 2018 November Q7
10 marks Standard +0.3
7 A particle moves in a straight line. The particle is initially at rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle is given by \(a = 25 - t ^ { 2 }\) for \(0 \leqslant t \leqslant 9\).
  1. Find the maximum velocity of the particle in this time period.
  2. Find the total distance travelled until the maximum velocity is reached.
    The acceleration of the particle for \(t > 9\) is given by \(a = - 3 t ^ { - \frac { 1 } { 2 } }\).
  3. Find the velocity of the particle when \(t = 25\).
    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 2019 November Q1
3 marks Easy -1.2
1 A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pulling force of 2500 N and the crate is moving at constant speed. Find the coefficient of friction between the crate and the ground.
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.
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 \(m\).