Questions — AQA M1 (172 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 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.
AQA M1 2013 June Q5
5 Two particles are connected by a light inextensible string that passes over a smooth peg. The particles have masses of 3 kg and 1 kg . The 1 kg particle is pulled down to ground level, where it is 40 cm below the level of the 3 kg particle, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-3_490_648_1272_696} The particles are released from rest with the string vertical above each particle. Assume that no resistance forces act on the particles as they move.
  1. By forming two equations of motion, one for each particle, find the magnitude of the acceleration of the particles after they have been released but before the 3 kg particle hits the ground.
  2. Find the speed of the 1 kg particle when the 3 kg particle hits the ground.
  3. After the 3 kg particle has hit the ground, the 1 kg particle continues to move and the string is now slack. Find the maximum height above ground level reached by the 1 kg particle.
  4. If a constant air resistance force also acts on the particles as they move, explain how this would change your answer for the acceleration in part (a). Give a reason for your answer.
AQA M1 2013 June Q6
6 In a scene from an action movie, a car is driven off the edge of a cliff and lands on the deck of a boat in the sea, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-4_355_1406_427_324} To land on the boat, the car must move 20 metres horizontally from the cliff. The level of the deck of the boat is 8 metres below the top of the cliff. Assume that the car is a particle which is travelling horizontally when it leaves the top of the cliff and that the car is not affected by air resistance as it moves.
  1. Find the time that it takes for the car to reach the deck of the boat.
  2. Find the speed at which the car is travelling when it leaves the top of the cliff.
  3. Find the speed of the car when it hits the deck of the boat.
AQA M1 2013 June Q7
7 A block of mass 30 kg is dragged across a rough horizontal surface by a rope that is at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the surface is 0.4 .
  1. The tension in the rope is 150 newtons.
    1. Draw a diagram to show the forces acting on the block as it moves.
    2. Show that the magnitude of the normal reaction force on the block is 243 newtons, correct to three significant figures.
    3. Find the magnitude of the friction force acting on the block.
    4. Find the acceleration of the block.
  2. When the block is moving, the tension is reduced so that the block moves at a constant speed, with the angle between the rope and the horizontal unchanged. Find the tension in the rope when the block is moving at this constant speed.
  3. If the block were made to move at a greater constant speed, again with the angle between the rope and the horizontal unchanged, how would the tension in this case compare to the tension found in part (b)?
AQA M1 2013 June Q8
8 A helicopter travels at a constant height above the sea. It passes directly over a lighthouse with position vector \(( 500 \mathbf { i } + 200 \mathbf { j } )\) metres relative to the origin, with a velocity of \(( - 17.5 \mathbf { i } - 27 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The helicopter moves with a constant acceleration of \(( 0.5 \mathbf { i } + 0.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find the position vector of the helicopter \(t\) seconds after it has passed over the lighthouse.
  2. The position vector of a rock is \(( 200 \mathbf { i } - 400 \mathbf { j } )\) metres relative to the origin. Show that the helicopter passes directly over the rock, and state the time that it takes for the helicopter to move from the lighthouse to the rock.
  3. Find the average velocity of the helicopter as it moves from the lighthouse to the rock.
  4. Is the magnitude of the average velocity equal to the average speed of the helicopter? Give a reason for your answer.
AQA M1 2014 June Q1
3 marks
1 A car is travelling along a straight horizontal road. It is moving at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it starts to accelerate. It accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 12 seconds.
  1. Find the speed of the car at the end of the 12 seconds.
  2. Find the distance travelled during the 12 seconds.
  3. The mass of the car is 1400 kg . A horizontal forward driving force of 1600 N acts on the car during the 12 seconds. Find the magnitude of the resistance force that acts on the car.
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-02_1513_1709_1192_153}
AQA M1 2014 June Q2
2 Three forces are in equilibrium in a vertical plane, as shown in the diagram. There is a vertical force of magnitude 40 N and a horizontal force of magnitude 60 N . The third force has magnitude \(F\) newtons and acts at an angle \(\theta\) above the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-04_490_894_456_571}
  1. \(\quad\) Find \(F\).
  2. \(\quad\) Find \(\theta\).
AQA M1 2014 June Q3
3 A skip, of mass 800 kg , is at rest on a rough horizontal surface. The coefficient of friction between the skip and the ground is 0.4 . A rope is attached to the skip and then the rope is pulled by a van so that the rope is horizontal while it is taut, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-06_237_1118_497_463} The mass of the van is 1700 kg . A constant horizontal forward driving force of magnitude \(P\) newtons acts on the van. The skip and the van accelerate at \(0.05 \mathrm {~ms} ^ { - 2 }\). Model both the van and the skip as particles connected by a light inextensible rope. Assume that there is no air resistance acting on the skip or on the van.
  1. Find the speed of the van and the skip when they have moved 6 metres.
  2. Draw a diagram to show the forces acting on the skip while it is accelerating.
  3. Draw a diagram to show the forces acting on the van while it is accelerating. State one advantage of modelling the van as a particle when considering the vertical forces.
  4. Find the magnitude of the friction force acting on the skip.
  5. Find the tension in the rope.
  6. \(\quad\) Find \(P\).
    \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-06_771_1703_1932_155}
AQA M1 2014 June Q4
4 A boat is crossing a river, which has two parallel banks. The width of the river is 20 metres. The water in the river is flowing at a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The boat sets off from the point \(O\) on one bank. The point \(A\) is directly opposite \(O\) on the other bank. The velocity of the boat relative to the water is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) to the bank. The boat lands at the point \(B\) which is 3 metres from \(A\). The angle between the actual path of the boat and the bank is \(\alpha ^ { \circ }\). The river and the velocities are shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-10_490_1307_641_370}
  1. Find the time that it takes for the boat to cross the river.
  2. Find \(\alpha\).
  3. \(\quad\) Find \(V\).
AQA M1 2014 June Q5
5 marks
5 Two particles, \(A\) and \(B\), have masses of \(m\) and \(k m\) respectively, where \(k\) is a constant. The particles are moving on a smooth horizontal plane when they collide and coalesce to form a single particle. Just before the collision the velocities of \(A\) and \(B\) are \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 6 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. Immediately after the collision the combined particle has velocity \(( 5.2 \mathbf { i } - 0.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find \(k\).
[0pt] [5 marks]