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 2009 January Q2
2 The graph shows how the velocity of a particle varies during a 50 -second period as it moves forwards and then backwards on a straight line.
\includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-2_615_1312_1007_301}
  1. State the times at which the velocity of the particle is zero.
  2. Show that the particle travels a distance of 75 metres during the first 30 seconds of its motion.
  3. Find the total distance travelled by the particle during the 50 seconds.
  4. Find the distance of the particle from its initial position at the end of the 50 -second period.
AQA M1 2009 January Q3
3 A box of mass 4 kg is held at rest on a plane inclined at an angle of \(40 ^ { \circ }\) to the horizontal. The box is then released and slides down the plane.
  1. A simple model assumes that the only forces acting on the box are its weight and the normal reaction from the plane. Show that, according to this simple model, the acceleration of the box would be \(6.30 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to three significant figures.
  2. In fact, the box moves down the plane with constant acceleration and travels 0.9 metres in 0.6 seconds. By using this information, find the acceleration of the box.
  3. Explain why the answer to part (b) is less than the answer to part (a).
AQA M1 2009 January Q4
4 Two particles, \(A\) and \(B\), are connected by a string that passes over a fixed peg, as shown in the diagram. The mass of \(A\) is 9 kg and the mass of \(B\) is 11 kg .
\includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-3_320_538_1117_806} The particles are released from rest in the position shown, where \(B\) is \(d\) metres higher than \(A\). The motion of the particles is to be modelled using simple assumptions.
  1. State one assumption that should be made about the peg.
  2. State two assumptions that should be made about the string.
  3. By forming an equation of motion for each of the particles \(A\) and \(B\), show that the acceleration of each particle has magnitude \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  4. When the particles have been moving for 0.5 seconds, they are at the same level.
    1. Find the speed of the particles at this time.
    2. Find \(d\).
AQA M1 2009 January Q5
5 A sledge of mass 8 kg is at rest on a rough horizontal surface. A child tries to move the sledge by pushing it with a pole, as shown in the diagram, but the sledge does not move. The pole is at an angle of \(30 ^ { \circ }\) to the horizontal and exerts a force of 40 newtons on the sledge.
\includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-4_221_922_513_552} Model the sledge as a particle.
  1. Draw a diagram to show the four forces acting on the sledge.
  2. Show that the normal reaction force between the sledge and the surface has magnitude 98.4 N .
  3. Find the magnitude of the friction force that acts on the sledge.
  4. Find the least possible value of the coefficient of friction between the sledge and the surface.
AQA M1 2009 January Q6
6 Two forces, \(\mathbf { P } = ( 6 \mathbf { i } - 3 \mathbf { j } )\) newtons and \(\mathbf { Q } = ( 3 \mathbf { i } + 15 \mathbf { j } )\) newtons, act on a particle. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular.
  1. Find the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  2. Calculate the magnitude of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  3. When these two forces act on the particle, it has an acceleration of \(( 1.5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find the mass of the particle.
  4. The particle was initially at rest at the origin.
    1. Find an expression for the position vector of the particle when the forces have been applied to the particle for \(t\) seconds.
    2. Find the distance of the particle from the origin when the forces have been applied to the particle for 2 seconds.
AQA M1 2009 January Q7
7 A boat is travelling in water that is moving north-east at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the boat relative to the water is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due west. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_275_349_415_504} \captionsetup{labelformat=empty} \caption{Velocity of the water}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_81_293_534_1181} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
\end{figure}
  1. Show that the magnitude of the resultant velocity of the boat is \(3.85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  2. Find the bearing on which the boat is travelling, giving your answer to the nearest degree.
AQA M1 2009 January Q8
8 A cricket ball is hit at ground level on a horizontal surface. It initially moves at \(28 \mathrm {~ms} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal.
  1. Find the maximum height of the ball during its flight.
  2. The ball is caught when it is at a height of 2 metres above ground level, as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_332_1070_1601_477} Show that the time that it takes for the ball to travel from the point where it was hit to the point where it was caught is 4.28 seconds, correct to three significant figures.
  3. Find the speed of the ball when it is caught.
AQA M1 2011 January Q1
1 A trolley, of mass 5 kg , is moving in a straight line on a smooth horizontal surface. It has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a stationary trolley, of mass \(m \mathrm {~kg}\). Immediately after the collision, the trolleys move together with velocity \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(m\).
(3 marks)
AQA M1 2011 January Q2
2 The graph shows how the velocity of a train varies as it moves along a straight railway line.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-04_574_1595_402_203}
  1. Find the total distance travelled by the train.
  2. Find the average speed of the train.
  3. Find the acceleration of the train during the first 10 seconds of its motion.
  4. The mass of the train is 200 tonnes. Find the magnitude of the resultant force acting on the train during the first 10 seconds of its motion.
AQA M1 2011 January Q3
3 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal tow bar, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-06_277_901_484_584} Assume that a constant resistance force of magnitude 200 newtons acts on the car and a constant resistance force of magnitude 300 newtons acts on the caravan. A constant driving force of magnitude \(P\) newtons acts on the car in the direction of motion. The car and caravan accelerate at \(0.8 \mathrm {~ms} ^ { - 2 }\).
    1. Find \(P\).
    2. Find the magnitude of the force in the tow bar that connects the car to the caravan.
    1. Find the time that it takes for the speed of the car and caravan to increase from \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    2. Find the distance that they travel in this time.
  3. Explain why the assumption that the resistance forces are constant is unrealistic.
    (1 mark)
AQA M1 2011 January Q4
4 A canoe is paddled across a river which has a width of 20 metres. The canoe moves from the point \(X\) on one bank of the river to the point \(Y\) on the other bank, so that its path is a straight line at an angle \(\alpha\) to the banks. The velocity of the canoe relative to the water is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the banks. The water flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-10_469_1333_543_374} Model the canoe as a particle.
  1. Find the magnitude of the resultant velocity of the canoe.
  2. Find the angle \(\alpha\).
  3. Find the time that it takes for the canoe to travel from \(X\) to \(Y\).
    \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-11_2486_1714_221_153}
AQA M1 2011 January Q5
5 A particle moves with constant acceleration \(( - 0.4 \mathbf { i } + 0.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially, it has velocity \(( 4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find an expression for the velocity of the particle at time \(t\) seconds.
    1. Find the velocity of the particle when \(t = 22.5\).
    2. State the direction in which the particle is travelling at this time.
  3. Find the time when the speed of the particle is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
AQA M1 2011 January Q6
6 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes over a smooth peg. Particle \(A\) has mass 2 kg and particle \(B\) has mass 4 kg . Particle \(A\) hangs freely with the string vertical. Particle \(B\) is at rest in equilibrium on a rough horizontal surface with the string at an angle of \(30 ^ { \circ }\) to the vertical. The particles, peg and string are shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-14_419_953_571_541}
  1. By considering particle \(A\), find the tension in the string.
  2. Draw a diagram to show the forces acting on particle \(B\).
  3. Show that the magnitude of the normal reaction force acting on particle \(B\) is 22.2 newtons, correct to three significant figures.
  4. Find the least possible value of the coefficient of friction between particle \(B\) and the surface.
    \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-16_2486_1714_221_153}
AQA M1 2011 January Q7
7 An arrow is fired from a point at a height of 1.5 metres above horizontal ground. It has an initial velocity of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. The arrow hits a target at a height of 1 metre above horizontal ground. The path of the arrow is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-18_341_1260_550_390} Model the arrow as a particle.
  1. Show that the time taken for the arrow to travel to the target is 1.30 seconds, correct to three significant figures.
  2. Find the horizontal distance between the point where the arrow is fired and the target.
  3. Find the speed of the arrow when it hits the target.
  4. Find the angle between the velocity of the arrow and the horizontal when the arrow hits the target.
  5. State one assumption that you have made about the forces acting on the arrow.
    (1 mark)
    \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-19_2486_1714_221_153}
    \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-20_2486_1714_221_153}
AQA M1 2011 January Q8
8 A van, of mass 2000 kg , is towed up a slope inclined at \(5 ^ { \circ }\) to the horizontal. The tow rope is at an angle of \(12 ^ { \circ }\) to the slope. The motion of the van is opposed by a resistance force of magnitude 500 newtons. The van is accelerating up the slope at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-22_269_991_513_529} Model the van as a particle.
  1. Draw a diagram to show the forces acting on the van.
  2. Show that the tension in the tow rope is 3480 newtons, correct to three significant figures.
AQA M1 2012 January Q1
1 Two particles, \(A\) of mass 7 kg and \(B\) of mass 3 kg , are moving on a smooth horizontal plane when they collide. Just before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). During the collision, the particles coalesce to form a single combined particle. Find the velocity of the single combined particle after the collision.
AQA M1 2012 January Q2
2 A block, of mass 4 kg , is made to move in a straight line on a rough horizontal surface by a horizontal force of 50 newtons, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-2_113_1075_913_486} Assume that there is no air resistance acting on the block.
  1. Draw a diagram to show all the forces acting on the block.
  2. Find the magnitude of the normal reaction force acting on the block.
  3. The acceleration of the block is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the friction force acting on the block.
  4. Find the coefficient of friction between the block and the surface.
  5. Explain how and why your answer to part (d) would change if you assumed that air resistance did act on the block.
AQA M1 2012 January Q3
3 The diagram shows a velocity-time graph for a train as it moves on a straight horizontal track for 50 seconds.
\includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-3_620_1221_408_358}
  1. Find the distance that the train moves in the first 28 seconds.
  2. Calculate the total distance moved by the train during the 50 seconds.
  3. Hence calculate the average speed of the train.
  4. Find the displacement of the train from its initial position when it has been moving for 50 seconds.
  5. Hence calculate the average velocity of the train.
  6. Find the acceleration of the train in the first 18 seconds of its motion.
AQA M1 2012 January Q4
4 A small ferry is used to cross a river which has straight parallel banks that are 200 metres apart. The water in the river moves at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ferry travels from a point \(A\) on one bank to a point \(B\) directly opposite \(A\) on the other bank. The velocity of the ferry relative to the water is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the upstream bank, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-4_467_1122_550_459}
  1. \(\quad\) Find \(V\).
  2. Find the time that it takes for the ferry to cross the river, giving your answer to the nearest second.
AQA M1 2012 January Q5
5 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal towbar. A resistance force of magnitude \(R\) newtons acts on the car and a resistance force of magnitude \(2 R\) newtons acts on the caravan. The car and caravan accelerate at a constant \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when a driving force of magnitude 4720 newtons acts on the car.
  1. Find \(R\).
  2. Find the tension in the towbar.
AQA M1 2012 January Q6
6 A cyclist freewheels, with a constant acceleration, in a straight line down a slope. As the cyclist moves 50 metres, his speed increases from \(4 \mathrm {~ms} ^ { - 1 }\) to \(10 \mathrm {~ms} ^ { - 1 }\).
    1. Find the acceleration of the cyclist.
    2. Find the time that it takes the cyclist to travel this distance.
  2. The cyclist has a mass of 70 kg . Calculate the magnitude of the resultant force acting on the cyclist.
  3. The slope is inclined at an angle \(\alpha\) to the horizontal.
    1. Find \(\alpha\) if it is assumed that there is no resistance force acting on the cyclist.
    2. Find \(\alpha\) if it is assumed that there is a constant resistance force of magnitude 30 newtons acting on the cyclist.
  4. Make a criticism of the assumption described in part (c)(ii).
AQA M1 2012 January Q7
7 A helicopter is initially at rest on the ground at the origin when it begins to accelerate in a vertical plane. Its acceleration is \(( 4.2 \mathbf { i } + 2.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) for the first 20 seconds of its motion. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively. Assume that the helicopter moves over horizontal ground.
  1. Find the height of the helicopter above the ground at the end of the 20 seconds.
  2. Find the velocity of the helicopter at the end of the 20 seconds.
  3. Find the speed of the helicopter when it is at a height of 180 metres above the ground.
AQA M1 2012 January Q8
8 A girl stands at the edge of a quay and sees a tin can floating in the water. The water level is 5 metres below the top of the quay and the can is at a horizontal distance of 10 metres from the quay, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-6_428_899_482_575} The girl decides to throw a stone at the can. She throws the stone from a height of 1 metre above the top of the quay. The initial velocity of the stone is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) below the horizontal, so that the initial velocity of the stone is directed at the can, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-6_437_903_1238_571} Assume that the stone is a particle and that it experiences no air resistance as it moves.
  1. Find \(\alpha\).
  2. Find the time that it takes for the stone to reach the level of the water.
  3. Find the distance between the stone and the can when the stone hits the water.
AQA M1 2013 January Q1
1 A car travels on a straight horizontal race track. The car decelerates uniformly from a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it travels a distance of 640 metres. The car then accelerates uniformly, travelling a further 1820 metres in 70 seconds.
    1. Find the time that it takes the car to travel the first 640 metres.
    2. Find the deceleration of the car during the first 640 metres.
    1. Find the acceleration of the car as it travels the further 1820 metres.
    2. Find the speed of the car when it has completed the further 1820 metres.
  3. Find the average speed of the car as it travels the 2460 metres.
AQA M1 2013 January Q2
2 Three forces act on a particle. These forces are ( \(9 \mathbf { i } - 3 \mathbf { j }\) ) newtons, ( \(5 \mathbf { i } + 8 \mathbf { j }\) ) newtons and ( \(- 7 \mathbf { i } + 3 \mathbf { j }\) ) newtons. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the resultant of these forces.
  2. Find the magnitude of the resultant force.
  3. Given that the particle has mass 5 kg , find the magnitude of the acceleration of the particle.
  4. Find the angle between the resultant force and the unit vector \(\mathbf { i }\).