Questions M1 (1912 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA M1 2010 June Q4
4 A particle, of mass \(m \mathrm {~kg}\), remains in equilibrium under the action of three forces, which act in a vertical plane, as shown in the diagram. The force with magnitude 60 N acts at \(48 ^ { \circ }\) above the horizontal and the force with magnitude 50 N acts at an angle \(\theta\) above the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-08_576_647_548_701}
  1. By resolving horizontally, find \(\theta\).
  2. Find \(m\).
    \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-09_2484_1709_223_153}
    \begin{center} \begin{tabular}{|l|l|} \hline & \begin{tabular}{l}
AQA M1 2010 June Q5
5 An aeroplane is travelling along a straight line between two points, \(A\) and \(B\), which are at the same height. The air is moving due east at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Relative to the air, the aeroplane travels due north at a speed of \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the resultant velocity of the aeroplane.
    (3 marks)
  2. Find the bearing on which the aeroplane is travelling, giving your answer to the nearest degree.
    (2 marks)
    \end{tabular}
    \hline QUESTION PART REFERENCE &
    \hline \end{tabular} \end{center}
    \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-11_2484_1709_223_153}
AQA M1 2010 June Q6
6 Two particles, \(A\) and \(B\), have masses 12 kg and 8 kg respectively. They are connected by a light inextensible string that passes over a smooth fixed peg, as shown in the diagram. $$A ( 12 \mathrm {~kg} )$$ The particles are released from rest and move vertically. Assume that there is no air resistance.
  1. By forming two equations of motion, show that the magnitude of the acceleration of each particle is \(1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the tension in the string.
  3. After the particles have been moving for 2 seconds, both particles are at a height of 4 metres above a horizontal surface. When the particles are in this position, the string breaks.
    1. Find the speed of particle \(A\) when the string breaks.
    2. Find the speed of particle \(A\) when it hits the surface.
    3. Find the time that it takes for particle \(B\) to reach the surface after the string breaks. Assume that particle \(B\) does not hit the peg.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-13_2484_1709_223_153}
AQA M1 2010 June Q7
7 A particle, of mass 10 kg , moves on a smooth horizontal surface. A single horizontal force, \(( 9 \mathbf { i } + 12 \mathbf { j } )\) newtons, acts on the particle. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find the acceleration of the particle.
  2. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the velocity of the particle is \(( 2.2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the particle is at the origin.
    1. Find the distance between the particle and the origin when \(t = 5\).
    2. Express \(\mathbf { v }\) in terms of \(t\).
    3. Find \(t\) when the particle is travelling north-east.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-15_2484_1709_223_153}
AQA M1 2010 June Q8
8 A ball is struck so that it leaves a horizontal surface travelling at \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The path of the ball is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-16_293_1364_461_347}
  1. Show that the ball takes \(\frac { 3 \sin \alpha } { 2 }\) seconds to reach its maximum height.
  2. The ball reaches a maximum height of 7 metres.
    1. Find \(\alpha\).
    2. Find the range, \(O A\).
  3. State two assumptions that you needed to make in order to answer the earlier parts of this question. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-17_2347_1691_223_153}
    \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-18_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-19_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-20_2505_1734_212_138}
AQA M1 2011 June Q1
1 A crane is used to lift a load, using a single vertical cable which is attached to the load. The load accelerates uniformly from rest. When it has risen 0.9 metres, its speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that the acceleration of the load is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    2. Find the time taken for the load to rise 0.9 metres.
  2. Given that the mass of the load is 800 kg , find the tension in the cable while the load is accelerating.
AQA M1 2011 June Q2
2 A wooden block, of mass 4 kg , is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.3 . A horizontal force, of magnitude 30 newtons, acts on the block and causes it to accelerate.
\includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-2_111_771_1146_639}
  1. Draw a diagram to show all the forces acting on the block.
  2. Calculate the magnitude of the normal reaction force acting on the block.
  3. Find the magnitude of the friction force acting on the block.
  4. Find the acceleration of the block.
AQA M1 2011 June Q3
3 A pair of cameras records the time that it takes a car on a motorway to travel a distance of 2000 metres. A car passes the first camera whilst travelling at \(32 \mathrm {~ms} ^ { - 1 }\). The car continues at this speed for 12.5 seconds and then decelerates uniformly until it passes the second camera when its speed has decreased to \(18 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the distance travelled by the car in the first 12.5 seconds.
  2. Find the time for which the car is decelerating.
  3. Sketch a speed-time graph for the car on this 2000-metre stretch of motorway.
  4. Find the average speed of the car on this 2000-metre stretch of motorway.
AQA M1 2011 June Q4
4 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface when they collide. The mass of \(A\) is 6 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Before the collision, the velocity of \(A\) is \(( 5 \mathbf { i } + 18 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). After the collision, the velocity of \(A\) is \(8 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) and the velocity of \(B\) is \(V \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
  1. Find \(m\).
  2. \(\quad\) Find \(V\).
AQA M1 2011 June Q5
5 Two particles, \(P\) and \(Q\), are connected by a string that passes over a fixed smooth peg, as shown in the diagram. The mass of \(P\) is 5 kg and the mass of \(Q\) is 3 kg .
\includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-3_209_433_1009_808} The particles are released from rest in the position shown.
  1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the tension in the string.
  3. State two modelling assumptions that you have made about the string.
  4. Particle \(P\) hits the floor when it has moved 0.196 metres and \(Q\) has not reached the peg.
    1. Find the time that it takes \(P\) to reach the floor.
    2. Find the speed of \(P\) when it hits the floor.
AQA M1 2011 June Q6
6 A bullet is fired horizontally from the top of a vertical cliff, at a height of \(h\) metres above the sea. It hits the sea 4 seconds after being fired, at a distance of 1000 metres from the base of the cliff, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-4_353_901_479_571}
  1. Find the initial speed of the bullet.
  2. \(\quad\) Find \(h\).
  3. Find the speed of the bullet when it hits the sea.
  4. Find the angle between the velocity of the bullet and the horizontal when it hits the sea.
AQA M1 2011 June Q7
7 A helicopter is initially hovering above a lighthouse. It then sets off so that its acceleration is \(( 0.5 \mathbf { i } + 0.375 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The helicopter does not change its height above sea level as it moves. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find the speed of the helicopter 20 seconds after it leaves its position above the lighthouse.
  2. Find the bearing on which the helicopter is travelling, giving your answer to the nearest degree.
  3. The helicopter stops accelerating when it is 500 metres from its initial position. Find the time that it takes for the helicopter to travel from its initial position to the point where it stops accelerating.
AQA M1 2011 June Q8
8 Three forces act in a vertical plane on an object of mass 250 kg , as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-5_481_1139_408_447} The two forces \(P\) newtons and \(Q\) newtons each act at \(80 ^ { \circ }\) to the horizontal. The object accelerates horizontally at \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) under the action of these forces.
  1. Show that $$P = 125 \left( \frac { a } { \cos 80 ^ { \circ } } + \frac { g } { \sin 80 ^ { \circ } } \right)$$
  2. Find the value of \(a\) for which \(Q\) is zero.
AQA M1 2012 June Q1
1 As a boat moves, it travels at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north, relative to the water. The water is moving due west at \(2 \mathrm {~ms} ^ { - 1 }\).
  1. Find the magnitude of the resultant velocity of the boat.
  2. Find the bearing of the resultant velocity of the boat.
AQA M1 2012 June Q2
2 Two toy trains, \(A\) and \(B\), are moving in the same direction on a straight horizontal track when they collide. As they collide, the speed of \(A\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, they move together with a speed of \(3.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(A\) is 2 kg . Find the mass of \(B\).
AQA M1 2012 June Q3
3 A car is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. The driver applies the brakes and a constant braking force acts on the car until it comes to rest.
  1. Assume that no other horizontal forces act on the car.
    1. After the car has travelled 75 metres, its speed has reduced to \(10 \mathrm {~ms} ^ { - 1 }\). Find the acceleration of the car.
    2. Find the time taken for the speed of the car to reduce from \(20 \mathrm {~ms} ^ { - 1 }\) to zero.
    3. Given that the mass of the car is 1400 kg , find the magnitude of the constant braking force.
  2. Given that a constant air resistance force of magnitude 200 N acts on the car during the motion, find the magnitude of the constant braking force.
    (1 mark)
AQA M1 2012 June Q4
4 A particle, of weight \(W\) newtons, is held in equilibrium by two forces of magnitudes 10 newtons and 20 newtons. The 10 -newton force is horizontal and the 20 -newton force acts at an angle \(\theta\) above the horizontal, as shown in the diagram. All three forces act in the same vertical plane.
\includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-3_406_608_520_717}
  1. \(\quad\) Find \(\theta\).
  2. \(\quad\) Find \(W\).
  3. Calculate the mass of the particle.
AQA M1 2012 June Q5
5 A block, of mass 12 kg , lies on a horizontal surface. The block is attached to a particle, of mass 18 kg , by a light inextensible string which passes over a smooth fixed peg. Initially, the block is held at rest so that the string supports the particle, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-3_346_716_1557_715} The block is then released.
  1. Assuming that the surface is smooth, use two equations of motion to find the magnitude of the acceleration of the block and particle.
  2. In reality, the surface is rough and the acceleration of the block is \(3 \mathrm {~ms} ^ { - 2 }\).
    1. Find the tension in the string.
    2. Calculate the magnitude of the normal reaction force acting on the block.
    3. Find the coefficient of friction between the block and the surface.
  3. State two modelling assumptions, other than those given, that you have made in answering this question.
AQA M1 2012 June Q6
6 A child pulls a sledge, of mass 8 kg , along a rough horizontal surface, using a light rope. The coefficient of friction between the sledge and the surface is 0.3 . The tension in the rope is \(T\) newtons. The rope is kept at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-4_273_775_516_644} Model the sledge as a particle.
  1. Draw a diagram to show all the forces acting on the sledge.
  2. Find the magnitude of the normal reaction force acting on the sledge, in terms of \(T\).
  3. Given that the sledge accelerates at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find \(T\).
AQA M1 2012 June Q7
7 A particle moves with a constant acceleration of \(( 0.1 \mathbf { i } - 0.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). It is initially at the origin where it has velocity \(( - \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find an expression for the position vector of the particle \(t\) seconds after it has left the origin.
  2. Find the time that it takes for the particle to reach the point where it is due east of the origin.
  3. Find the speed of the particle when it is travelling south-east.
AQA M1 2012 June Q8
8 A particle is launched from the point \(A\) on a horizontal surface, with a velocity of \(22.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-5_369_1182_406_431} After 2 seconds, the particle reaches the point \(C\), where it is at its maximum height above the surface.
  1. Show that \(\sin \theta = 0.875\).
  2. Find the height of the point \(C\) above the horizontal surface.
  3. The particle returns to the surface at the point \(B\). Find the distance between \(A\) and \(B\). (3 marks)
  4. Find the length of time during which the height of the particle above the surface is greater than 5 metres.
  5. Find the minimum speed of the particle.
AQA M1 2013 June Q1
1 A toy train of mass 300 grams is moving along a straight horizontal track at a speed of \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This toy train collides with another toy train, of mass 200 grams, which is at rest on the same track. During the collision, the two trains lock together and then move together. Find the speed of the trains immediately after the collision.
AQA M1 2013 June Q2
2 The graph shows how the speed of a cyclist, Hannah, varies as she travels for 21 seconds along a straight horizontal road.
\includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-2_590_1603_847_230}
  1. Find the distance travelled by Hannah in the 21 seconds.
  2. Find Hannah's average speed during the 21 seconds.
AQA M1 2013 June Q3
3 A ship travels through water that is moving due east at a speed of \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ship travels due north relative to the water at a speed of \(7 \mathrm {~ms} ^ { - 1 }\). The resultant velocity of the ship is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a bearing \(\alpha\). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Velocity of the
water} \includegraphics[alt={},max width=\textwidth]{cb5001b1-1744-439f-aa35-8dd01bc90421-2_387_391_2069_653}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Velocity of the ship relative to the water} \includegraphics[alt={},max width=\textwidth]{cb5001b1-1744-439f-aa35-8dd01bc90421-2_214_167_2165_1334}
\end{figure}
  1. \(\quad\) Find \(V\).
  2. Find \(\alpha\), giving your answer as a three-figure bearing, correct to the nearest degree.
AQA M1 2013 June Q4
4 Two forces, acting at a point, have magnitudes of 40 newtons and 70 newtons. The angle between the two forces is \(30 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-3_213_531_400_759}
  1. Find the magnitude of the resultant of these two forces.
  2. Find the angle between the resultant force and the 70 newton force.