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AQA M1 2009 June Q1
5 marks Moderate -0.5
1 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface when they collide. During the collision, the two particles coalesce to form a single combined particle. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { r } 6 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } - 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the velocity of the combined particle after the collision.
  2. Find the speed of the combined particle after the collision.
AQA M1 2009 June Q2
6 marks Moderate -0.8
2 A lift is travelling upwards and accelerating uniformly. During a 5 second period, it travels 16 metres and the speed of the lift increases from \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. \(\quad\) Find \(u\).
  2. Find the acceleration of the lift.
AQA M1 2009 June Q3
4 marks Easy -1.3
3 A car is travelling in a straight line on a horizontal road. A driving force, of magnitude 3000 N , acts in the direction of motion and a resistance force, of magnitude 600 N , opposes the motion of the car. Assume that no other horizontal forces act on the car.
  1. Find the magnitude of the resultant force on the car.
  2. The mass of the car is 1200 kg . Find the acceleration of the car. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-07_38_118_440_159}
    \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-07_40_118_529_159}
AQA M1 2009 June Q4
7 marks Moderate -0.8
4 A river has parallel banks which are 16 metres apart. The water in the river flows at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. A boat sets off from one bank at the point \(A\) and travels perpendicular to the bank so that it reaches the point \(B\), which is directly opposite the point \(A\). It takes the boat 10 seconds to cross the river. The velocity of the boat relative to the water has magnitude \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is at an angle \(\alpha\) to the bank, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-08_400_1011_667_511}
  1. Show that the magnitude of the resultant velocity of the boat is \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. \(\quad\) Find \(V\).
  3. Find \(\alpha\).
  4. State one modelling assumption that you needed to make about the boat.
    \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-08_72_1689_1617_154}
    .......... \(\_\_\_\_\)
    \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-09_40_118_529_159}
AQA M1 2009 June Q5
16 marks Moderate -0.3
5 A block, of mass 14 kg , is held at rest on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.25 . A light inextensible string, which passes over a fixed smooth peg, is attached to the block. The other end of the string is attached to a particle, of mass 6 kg , which is hanging at rest.
\includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-10_264_716_502_708} The block is released and begins to accelerate.
  1. Find the magnitude of the friction force acting on the block.
  2. By forming two equations of motion, one for the block and one for the particle, show that the magnitude of the acceleration of the block and the particle is \(1.225 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the tension in the string.
  4. When the block is released, it is 0.8 metres from the peg. Find the speed of the block when it hits the peg.
  5. When the block reaches the peg, the string breaks and the particle falls a further 0.5 metres to the ground. Find the speed of the particle when it hits the ground.
    (3 marks)
    \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-11_2484_1709_223_153}
AQA M1 2009 June Q6
13 marks Moderate -0.8
6 A ball is kicked from the point \(P\) on a horizontal surface. It leaves the surface with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal and hits the surface for the first time at the point \(Q\). Assume that the ball is a particle that moves only under the influence of gravity.
\includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-12_317_1118_513_461}
  1. Show that the time that it takes the ball to travel from \(P\) to \(Q\) is 3.13 s , correct to three significant figures.
  2. Find the distance between the points \(P\) and \(Q\).
  3. If a heavier ball were projected from \(P\) with the same velocity, how would the distance between \(P\) and \(Q\), calculated using the same modelling assumptions, compare with your answer to part (b)? Give a reason for your answer.
  4. Find the maximum height of the ball above the horizontal surface.
  5. State the magnitude and direction of the velocity of the ball as it hits the surface.
    \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-13_2484_1709_223_153}
AQA M1 2009 June Q7
12 marks Moderate -0.8
7 A particle moves on a smooth horizontal plane. It is initially at the point \(A\), with position vector \(( 9 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m }\), and has velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves with a constant acceleration of \(( 0.25 \mathbf { i } + 0.3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) for 20 seconds until it reaches the point \(B\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find the velocity of the particle at the point \(B\).
  2. Find the velocity of the particle when it is travelling due north.
  3. Find the position vector of the point \(B\).
  4. Find the average velocity of the particle as it moves from \(A\) to \(B\).
    \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-15_2484_1709_223_153}
AQA M1 2009 June Q8
12 marks Moderate -0.3
8 The diagram shows a block, of mass 20 kg , being pulled along a rough horizontal surface by a rope inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-16_323_1194_411_424} The coefficient of friction between the block and the surface is \(\mu\). Model the block as a particle which slides on the surface.
  1. If the tension in the rope is 60 newtons, the block moves at a constant speed.
    1. Show that the magnitude of the normal reaction force acting on the block is 166 N .
    2. Find \(\mu\).
  2. If the rope remains at the same angle and the block accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the tension in the rope. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-18_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-19_2488_1719_219_150}
AQA M1 2010 June Q1
9 marks Easy -1.2
1 A bus slows down as it approaches a bus stop. It stops at the bus stop and remains at rest for a short time as the passengers get on. It then accelerates away from the bus stop. The graph shows how the velocity of the bus varies.
\includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-02_627_1296_657_402} Assume that the bus travels in a straight line during the motion described by the graph.
  1. State the length of time for which the bus is at rest.
  2. Find the distance travelled by the bus in the first 40 seconds.
  3. Find the total distance travelled by the bus in the 120 -second period.
  4. Find the average speed of the bus in the 120 -second period.
  5. If the bus had not stopped but had travelled at a constant \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the 120 -second period, how much further would it have travelled?
    \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-03_2484_1709_223_153}
AQA M1 2010 June Q2
7 marks Moderate -0.8
2 A block, of mass 10 kg , is at rest on a rough horizontal surface, when a horizontal force, of magnitude \(P\) newtons, is applied to the block, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-04_108_962_461_539} The coefficient of friction between the block and the surface is 0.5 .
  1. Draw and label a diagram to show all the forces acting on the block.
    1. Calculate the magnitude of the normal reaction force acting on the block.
    2. Find the maximum possible magnitude of the friction force between the block and the surface.
    3. Given that \(P = 30\), state the magnitude of the friction force acting on the block.
  3. Given that \(P = 80\), find the acceleration of the block.
    \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-05_2484_1709_223_153}
AQA M1 2010 June Q3
6 marks Moderate -0.8
3 Two particles, \(A\) and \(B\), are moving on a smooth horizontal plane 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 \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). After the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { l } 7 \\ b \end{array} \right] \mathrm { ms } ^ { - 1 }\).
  1. Find \(m\).
  2. \(\quad\) Find \(b\).
    (2 marks)
    .......... \(\_\_\_\_\)
    \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_40_118_529_159}
    \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_39_117_623_159}
AQA M1 2010 June Q4
7 marks Moderate -0.8
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 marks Moderate -0.8
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
17 marks Moderate -0.3
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
11 marks Moderate -0.3
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
13 marks Moderate -0.8
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
8 marks Moderate -0.8
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
7 marks Moderate -0.8
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
9 marks Easy -1.2
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
6 marks Moderate -0.3
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
14 marks Standard +0.3
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
11 marks Moderate -0.8
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
12 marks Standard +0.3
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 marks Standard +0.3
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
5 marks Moderate -0.8
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.