Questions — AQA M1 (171 questions)

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AQA M1 2014 June Q6
8 marks Standard +0.3
6 A bullet is fired from a rifle at a target, which is at a distance of 420 metres from the rifle. The bullet leaves the rifle travelling at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at an angle of \(2 ^ { \circ }\) above the horizontal. The centre of the target, \(C\), is at the same horizontal level as the rifle. The bullet hits the target at the point \(A\), which is on a vertical line through \(C\). The bullet takes 1.8 seconds to reach the point \(A\).
  1. Find \(V\), showing clearly how you obtain your answer.
  2. Find the distance between \(A\) and \(C\).
  3. State one assumption that you have made about the forces acting on the bullet.
    [0pt] [1 mark]
AQA M1 2014 June Q7
11 marks Standard +0.3
7 Two particles, \(A\) and \(B\), move on a horizontal surface with constant accelerations of \(- 0.4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. At time \(t = 0\), particle \(A\) starts at the origin with velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\), particle \(B\) starts at the point with position vector \(11.2 \mathbf { i }\) metres, with velocity \(( 0.4 \mathbf { i } + 0.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the position vector of \(A , 10\) seconds after it leaves the origin.
    [0pt] [2 marks]
  2. Show that the two particles collide, and find the position vector of the point where they collide.
    [0pt] [9 marks]
    \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-16_1881_1707_822_153}
    \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-17_2484_1707_221_153}
AQA M1 2014 June Q8
12 marks Standard +0.3
8 A crate, of mass 40 kg , is initially at rest on a rough slope inclined at \(30 ^ { \circ }\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-18_355_882_411_587} The coefficient of friction between the crate and the slope is \(\mu\).
  1. Given that the crate is on the point of slipping down the slope, find \(\mu\).
  2. A horizontal force of magnitude \(X\) newtons is now applied to the crate, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-18_357_881_1208_575}
    1. Find the normal reaction on the crate in terms of \(X\).
    2. Given that the crate accelerates up the slope at \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find \(X\).
      [0pt] [5 marks]
      \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-19_2484_1707_221_153}
AQA M1 2015 June Q1
3 marks Easy -1.2
1 A child, of mass 48 kg , is initially standing at rest on a stationary skateboard. The child jumps off the skateboard and initially moves horizontally with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The skateboard moves with a speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the opposite direction to the direction of motion of the child. Find the mass of the skateboard.
[0pt] [3 marks]
AQA M1 2015 June Q2
5 marks Moderate -0.3
2 A yacht is sailing through water that is flowing due west at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the yacht relative to the water is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due south. The yacht has a resultant velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a bearing of \(\theta\).
  1. \(\quad\) Find \(V\).
  2. Find \(\theta\), giving your answer to the nearest degree.
AQA M1 2015 June Q3
7 marks Moderate -0.3
3 A ship has a mass of 500 tonnes. Two tugs are used to pull the ship using cables that are horizontal. One tug exerts a force of 100000 N at an angle of \(25 ^ { \circ }\) to the centre line of the ship. The other tug exerts a force of \(T \mathrm {~N}\) at an angle of \(20 ^ { \circ }\) to the centre line of the ship. The diagram shows the ship and forces as viewed from above. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-06_279_844_539_664} The ship accelerates in a straight line along its centre line.
  1. \(\quad\) Find \(T\).
  2. A resistance force of magnitude 20000 N directly opposes the motion of the ship. Find the acceleration of the ship.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-06_1419_1714_1288_153}
AQA M1 2015 June Q4
10 marks Moderate -0.8
4 A particle moves with constant acceleration between the points \(A\) and \(B\). At \(A\), it has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At \(B\), it has velocity \(( 7 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It takes 10 seconds to move from \(A\) to \(B\).
  1. Find the acceleration of the particle.
  2. Find the distance between \(A\) and \(B\).
  3. Find the average velocity as the particle moves from \(A\) to \(B\).
AQA M1 2015 June Q5
16 marks Standard +0.3
5 A block, of mass \(3 m\), is placed on a horizontal surface at a point \(A\). A light inextensible string is attached to the block and passes over a smooth peg. The string is horizontal between the block and the peg. A particle, of mass \(2 m\), is attached to the other end of the string. The block is released from rest with the string taut and the string between the peg and the particle vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-10_170_726_536_657} Assume that there is no air resistance acting on either the block or the particle, and that the size of the block is negligible. The horizontal surface is smooth between the points \(A\) and \(B\), but rough between the points \(B\) and \(C\). Between \(B\) and \(C\), the coefficient of friction between the block and the surface is 0.8 .
  1. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(A\) and \(B\).
  2. Given that the distance between the points \(A\) and \(B\) is 1.2 metres, find the speed of the block when it reaches \(B\).
  3. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(B\) and \(C\).
  4. Given that the distance between the points \(B\) and \(C\) is 0.9 metres, find the speed of the block when it reaches \(C\).
  5. Explain why it is important to assume that the size of the block is negligible.
    [0pt] [1 mark]
AQA M1 2015 June Q6
12 marks Standard +0.3
6 Emma is in a park with her dog, Roxy. Emma throws a ball and Roxy catches it in her mouth. The ground in the park is horizontal. Emma throws the ball from a point at a height of 1.2 metres above the ground and Roxy catches the ball when it is at a height of 0.5 metres above the ground. Emma throws the ball with an initial velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal.
  1. Find the time that the ball takes to travel from Emma's hand to Roxy's mouth.
  2. Find the horizontal distance travelled by the ball during its flight.
  3. During the flight, the speed of the ball is a maximum when it is at a height of \(h\) metres above the ground. Write down the value of \(h\).
  4. Find the maximum speed of the ball during its flight.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-14_1566_1707_1137_153}
AQA M1 2015 June Q7
11 marks Standard +0.3
7 Two forces, which act in a vertical plane, are applied to a crate. The crate has mass 50 kg , and is initially at rest on a rough horizontal surface. One force has magnitude 80 N and acts at an angle of \(30 ^ { \circ }\) to the horizontal and the other has magnitude 40 N and acts at an angle of \(20 ^ { \circ }\) to the horizontal. The forces are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-16_241_999_493_523} The coefficient of friction between the crate and the surface is 0.6 . Model the crate as a particle.
  1. Draw a diagram to show the forces acting on the crate.
  2. Find the magnitude of the normal reaction force acting on the crate.
  3. Does the crate start to move when the two forces are applied to the crate? Show all your working.
  4. State one aspect of the possible motion of the crate that is ignored by modelling it as a particle.
    [0pt] [1 mark]
AQA M1 2015 June Q8
11 marks Standard +0.3
8 Two trains, \(A\) and \(B\), are moving on straight horizontal tracks which run alongside each other and are parallel. The trains both move with constant acceleration. At time \(t = 0\), the fronts of the trains pass a signal. The velocities of the trains are shown in the graph below. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-18_633_1077_475_424}
  1. Find the distance between the fronts of the two trains when they have the same velocity and state which train has travelled further from the signal.
  2. Find the time when \(A\) has travelled 9 metres further than \(B\).
    \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-20_2288_1707_221_153}
AQA M1 2016 June Q2
3 marks Moderate -0.8
2 Three forces \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { N } , ( p \mathbf { i } + 5 \mathbf { j } ) \mathrm { N }\) and \(( - 8 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\) act on a particle of mass 5 kg to produce an acceleration of \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). No other forces act on the particle.
  1. Find the resultant force acting on the particle in terms of \(p\) and \(q\).
  2. \(\quad\) Find \(p\) and \(q\).
  3. Given that the particle is initially at rest, find the displacement of the particle from its initial position when these forces have been acting for 4 seconds.
    [0pt] [3 marks]
AQA M1 2016 June Q3
4 marks Moderate -0.8
3 A toy car is placed at the top of a ramp. After the car has been released from rest, it travels a distance of 1.08 metres down the ramp, in a time of 1.2 seconds. Assume that there is no resistance to the motion of the car.
  1. Find the magnitude of the acceleration of the car while it is moving down the ramp.
  2. Find the speed of the car, when it has travelled 1.08 metres down the ramp.
  3. Find the angle between the ramp and the horizontal, giving your answer to the nearest degree.
    [0pt] [4 marks]
AQA M1 2016 June Q4
3 marks Moderate -0.8
4 An aeroplane is flying in air that is moving due east at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Relative to the air, the aeroplane has a velocity of \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. During a 20 second period, the motion of the air causes the aeroplane to move 240 metres to the east.
  1. \(\quad\) Find \(V\).
  2. Find the magnitude of the resultant velocity of the aeroplane.
  3. Find the direction of the resultant velocity, giving your answer as a three-figure bearing, correct to the nearest degree.
    [0pt] [3 marks]
AQA M1 2016 June Q5
4 marks Moderate -0.3
5 Two particles, of masses 3 kg and 7 kg , are connected by a light inextensible string that passes over a smooth peg. The 3 kg particle is held at ground level with the string above it taut and vertical. The 7 kg particle is at a height of 80 cm above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-10_469_600_486_721} The 3 kg particle is then released from rest.
  1. By forming two equations of motion, show that the magnitude of the acceleration of the particles is \(3.92 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the speed of the 7 kg particle just before it hits the ground.
  3. When the 7 kg particle hits the ground, the string becomes slack and in the subsequent motion the 3 kg particle does not hit the peg. Find the maximum height of the 3 kg particle above the ground.
    [0pt] [4 marks]
AQA M1 2016 June Q6
6 marks Standard +0.3
6 A floor polisher consists of a heavy metal block with a polishing cloth attached to the underside. A light rod is also attached to the block and is used to push the block across the floor that is to be polished. The block has mass 5 kg . Assume that the floor is horizontal. Model the block as a particle. The coefficient of friction between the cloth and the floor is 0.2 .
A person pushes the rod to exert a force on the block. The force is at an angle of \(60 ^ { \circ }\) to the horizontal and the block accelerates at \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The diagram shows the block and the force exerted by the rod in this situation. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-14_309_205_772_1009} The rod exerts a force of magnitude \(T\) newtons on the block.
  1. Find, in terms of \(T\), the magnitude of the normal reaction force acting on the block.
  2. \(\quad\) Find \(T\).
    [0pt] [6 marks]
AQA M1 2016 June Q7
11 marks Moderate -0.3
7 At a school fair, there is a competition in which people try to kick a football so that it lands in a large rectangular box. The height of the top of the box is 1 metre and its width is 3 metres. One student kicks a football so that it initially moves at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. It hits the top front edge of the box, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-16_465_1342_625_351} Model the football as a particle that is not subject to any resistance forces as it moves.
  1. Find the time taken for the football to move from the point where it was kicked to the box.
  2. Find the horizontal distance from the point where the football is kicked to the front of the box.
  3. If the same student kicks the football at the same angle from the same initial position, what is the speed at which the student should kick the football if it is to hit the top back edge of the box?
  4. Explain the significance of modelling the football as a particle in this context.
    [0pt] [1 mark]
    \includegraphics[max width=\textwidth, alt={}]{5dd17095-18a6-470b-a24a-4456c8e3ed31-23_2488_1709_219_153}
    \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
AQA M1 Q1
Moderate -0.8
1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\). \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_147_506_644_733}
  1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
  2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_138_506_1144_730}
AQA M1 Q4
Standard +0.3
4 Water flows in a constant direction at a constant speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat travels in the water at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water.
  1. The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(74 ^ { \circ }\) to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
    \end{figure}
    1. Find \(V\).
    2. Show that \(u = 3.44\), correct to three significant figures.
  2. The boat changes course so that it travels relative to the water at an angle of \(45 ^ { \circ }\) to the direction of motion of the water. The resultant velocity of the boat is now of magnitude \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
    \includegraphics[max width=\textwidth, alt={}]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_132_273_1493_895}
    Velocity of the boat relative to the water \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_232_355_1498_1384} Find the value of \(v\).
    (4 marks)
AQA M1 Q5
Moderate -0.8
5 A golf ball is projected from a point \(O\) with initial velocity \(V\) at an angle \(\alpha\) to the horizontal. The ball first hits the ground at a point \(A\) which is at the same horizontal level as \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-005_227_602_484_735} It is given that \(V \cos \alpha = 6 u\) and \(V \sin \alpha = 2.5 u\).
  1. Show that the time taken for the ball to travel from \(O\) to \(A\) is \(\frac { 5 u } { g }\).
  2. Find, in terms of \(g\) and \(u\), the distance \(O A\).
  3. Find \(V\), in terms of \(u\).
  4. State, in terms of \(u\), the least speed of the ball during its flight from \(O\) to \(A\).
AQA M1 Q6
Moderate -0.8
6 A van moves from rest on a straight horizontal road.
  1. In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    During the second stage, the van accelerates from \(4 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
    During the third stage, the van accelerates from \(12 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\).
    1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
    2. Find the total distance that the van travels during the 30 seconds.
    3. Find the average speed of the van during the 30 seconds.
    4. Find the greatest acceleration of the van during the 30 seconds.
  2. In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-006_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
    1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
    2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).
AQA M1 Q7
Moderate -0.8
7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-007_360_296_502_904} The buckets are each of mass 0.6 kg .
    1. State the magnitude of the tension in the rope.
    2. State the magnitude and direction of the force exerted on the beam by the rope.
  2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
    1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
    2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.
AQA M1 2006 January Q1
6 marks Moderate -0.8
1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\). \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_147_506_644_733}
  1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
  2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_138_506_1144_730}
AQA M1 2006 January Q2
5 marks Moderate -0.8
2 A particle \(P\) moves with acceleration \(( - 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the velocity of \(P\) at time \(t\) seconds.
  2. Find the speed of \(P\) when \(t = 0.5\).
AQA M1 2006 January Q3
6 marks Easy -1.2
3
  1. A small stone is dropped from a height of 25 metres above the ground.
    1. Find the time taken for the stone to reach the ground.
    2. Find the speed of the stone as it reaches the ground.
  2. A large package is dropped from the same height as the stone. Explain briefly why the time taken for the package to reach the ground is likely to be different from that for the stone.
    (2 marks)