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 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 \(\mathbf { i }\).
AQA M1 2013 January Q3
3 A box, of mass 3 kg , is placed on a rough slope inclined at an angle of \(40 ^ { \circ }\) to the horizontal. It is released from rest and slides down the slope.
  1. Draw a diagram to show the forces acting on the box.
  2. Find the magnitude of the normal reaction force acting on the box.
  3. The coefficient of friction between the box and the slope is 0.2 . Find the magnitude of the friction force acting on the box.
  4. Find the acceleration of the box.
  5. State an assumption that you have made about the forces acting on the box.
AQA M1 2013 January Q4
4 A tractor, of mass 3500 kg , is used to tow a trailer, of mass 2400 kg , across a horizontal field. The trailer is connected to the tractor by a horizontal tow bar. As they move, a constant resistance force of 800 newtons acts on the trailer and a constant resistance force of \(R\) newtons acts on the tractor. A forward driving force of 2500 newtons acts on the tractor. The trailer and tractor accelerate at \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. \(\quad\) Find \(R\).
  2. Find the magnitude of the force that the tow bar exerts on the trailer.
  3. State the magnitude of the force that the tow bar exerts on the tractor.
AQA M1 2013 January Q5
5 Two particles, \(A\) and \(B\), are moving towards each other along the same straight horizontal line when they collide. Particle \(A\) has mass 5 kg and particle \(B\) has mass 4 kg . Just before the collision, the speed of \(A\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision, the speed of \(A\) is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and both particles move on the same straight horizontal line. Find the two possible speeds of \(B\) after the collision.
(6 marks)
AQA M1 2013 January Q6
6 A river has straight parallel banks. The water in the river is flowing at a constant velocity of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. A boat crosses the river, from the point \(A\) to the point \(B\), so that its path is at an angle \(\alpha\) to the bank. The velocity of the boat relative to the water is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the bank. The diagram shows these velocities and the path of the boat.
\includegraphics[max width=\textwidth, alt={}, center]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-12_467_988_568_532}
  1. Show that \(\alpha = 53.1 ^ { \circ }\), correct to three significant figures.
  2. The boat returns along the same straight path from \(B\) to \(A\). Given that the speed of the boat relative to the water is still \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the magnitude of the resultant velocity of the boat on the return journey.
AQA M1 2013 January Q7
7 A particle is initially at the point \(A\), which has position vector \(13.6 \mathbf { i }\) metres, with respect to an origin \(O\). At the point \(A\), the particle has velocity \(( 6 \mathbf { i } + 2.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and in its subsequent motion, it has a constant acceleration of \(( - 0.8 \mathbf { i } + 0.1 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find an expression for the velocity of the particle \(t\) seconds after it leaves \(A\).
  2. Find an expression for the position vector of the particle, with respect to the origin \(O\), \(t\) seconds after it leaves \(A\).
  3. Find the distance of the particle from the origin \(O\) when it is travelling in a north-westerly direction.
    \includegraphics[max width=\textwidth, alt={}]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-17_2486_1709_221_153}
AQA M1 2013 January Q8
8 A golf ball is hit from a point on a horizontal surface, so that it has an initial velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The ball travels through the air and after 2.4 seconds hits a vertical wall at a height of 3 metres. The wall is at a horizontal distance of 38.4 metres from the point where the ball was hit. The path of the ball is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-18_300_1000_566_520} Assume that the weight of the ball is the only force that acts on it as it travels through the air.
  1. Find the horizontal component of the velocity of the ball.
  2. \(\quad\) Find \(V\).
  3. \(\quad\) Find \(\alpha\).
AQA M1 2005 June Q1
1 A particle of mass \(m\) has velocity \(\left[ \begin{array} { l } 4
2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides with a particle of mass 3 kg which has velocity \(\left[ \begin{array} { l } - 1
- 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). During the collision the particles coalesce and move with velocity \(\left[ \begin{array} { l } 1
V \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Show that \(m = 2\).
  2. Find \(V\).
AQA M1 2005 June Q2
2 A train travels along a straight horizontal track between two points \(A\) and \(B\).
Initially the train is at \(A\) and moving at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Due to a problem, the train has to slow down and stop. At time \(t = 40\) seconds it begins to move again. At time \(t = 120\) seconds the train is at \(B\) and moving at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) again. The graph below shows how the velocity of the train varies as it moves from \(A\) to \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-2_408_1086_1505_434}
  1. Use the graph to find the total distance between the points \(A\) and \(B\).
  2. The train should have travelled between \(A\) and \(B\) at a constant velocity of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Calculate the time that the train would take to travel between \(A\) and \(B\) at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    2. Calculate the time by which the train was delayed.
  3. The train has mass 500 tonnes. Find the resultant force acting on the train when \(40 < t < 120\).
    (4 marks)
AQA M1 2005 June Q3
3 A boat can travel at a speed of \(2 \mathrm {~ms} ^ { - 1 }\) in still water. The boat is to cross a river in which a current flows at a speed of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angle between the direction in which the boat is pointing and the bank is \(\alpha\). The boat travels so that the resultant velocity of the boat is perpendicular to the bank.
\includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-3_264_1040_575_493}
  1. Show that \(\alpha = 66.4 ^ { \circ }\) correct to three significant figures.
    1. Find the magnitude of the resultant velocity of the boat.
    2. The width of the river is 14 metres. Find the time that it takes for the boat to cross the river.
AQA M1 2005 June Q4
4 Two particles, \(A\) of mass 5 kg and \(B\) of mass 9 kg , are attached to the ends of a light inextensible string. The string passes over a light smooth pulley as shown in the diagram. The particles are released from rest at the same height.
\includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-3_378_287_1580_872}
  1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the tension in the string.
  3. When \(B\) has been moving for 0.5 seconds it hits the floor. Find the height of \(A\), above the floor, at this time. Assume that \(A\) is still below the pulley when \(B\) hits the floor.
    (4 marks)
AQA M1 2005 June Q5
5 A sphere of mass 200 grams is released from rest and allowed to fall vertically.
  1. A student states that the acceleration of the sphere is \(9.8 \mathrm {~ms} ^ { - 2 }\) while it is falling. What modelling assumption is this student making?
  2. The student conducts an experiment and finds that the acceleration of the ball is in fact \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). He formulates a model for the motion that assumes a constant resistance force acts on the ball as it is falling.
    1. Calculate the magnitude of this resistance force based on this assumption.
    2. Describe how the resistance force would vary in reality.
  3. In a revised model the resistance force is assumed to be proportional to the speed of the sphere.
    1. State the initial acceleration of the sphere.
    2. State what would happen to the acceleration of the sphere if it were able to fall for a long period of time.
AQA M1 2005 June Q6
6 A ball is hit from horizontal ground with velocity \(( 10 \mathbf { i } + 24.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively.
  1. State two assumptions that you should make about the ball in order to make predictions about its motion.
  2. The path of the ball is shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_771_705_625}
    1. Show that the time of flight of the ball is 5 seconds.
    2. Find the range of the ball.
  3. In fact the ball hits a vertical wall that is 20 metres from the initial position of the ball.
    \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_403_1466_769} Find the height of the ball when it hits the wall.
  4. If a heavier ball were projected in the same way, would your answers to part (b) of this question change? Explain why.
AQA M1 2005 June Q7
7 A particle moves on a smooth horizontal surface with acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). Initially the velocity of the particle is \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find an expression for the velocity of the particle at time \(t\) seconds.
  2. Find the time when the particle is travelling in the \(\mathbf { i }\) direction.
  3. Show that when \(t = 4\) the speed of the particle is \(20 \mathrm {~ms} ^ { - 1 }\).
AQA M1 2005 June Q8
8 A rough slope is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. A particle of mass 6 kg is on the slope. A string is attached to the particle and is at an angle of \(30 ^ { \circ }\) to the slope. The tension in the string is 20 N . The diagram shows the slope, the particle and the string.
\includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-6_259_684_518_676} The particle moves up the slope with an acceleration of \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Draw a diagram to show the forces acting on the particle.
  2. Show that the magnitude of the normal reaction force is 47.9 N , correct to three significant figures.
  3. Find the coefficient of friction between the particle and the slope.
AQA M1 2006 June Q1
1 A stone is dropped from a high bridge and falls vertically.
  1. Find the distance that the stone falls during the first 4 seconds of its motion.
  2. Find the average speed of the stone during the first 4 seconds of its motion.
  3. State one modelling assumption that you have made about the forces acting on the stone during the motion.
AQA M1 2006 June Q2
2 A particle is in equilibrium under the action of four horizontal forces of magnitudes 5 newtons, 8 newtons, \(P\) newtons and \(Q\) newtons, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-2_355_357_1146_852}
  1. Show that \(P = 9\).
  2. Find the value of \(Q\).