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AQA M2 2006 June Q6
7 marks Moderate -0.8
6 A car of mass 1200 kg travels round a roundabout on a horizontal, circular path at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The radius of the circle is 50 metres. Assume that there is no resistance to the motion of the car and that the car can be modelled as a particle.
  1. A friction force, directed towards the centre of the roundabout, acts on the car as it moves. Show that the magnitude of this friction force is 4704 N .
  2. The coefficient of friction between the car and the road is \(\mu\). Show that \(\mu \geqslant 0.4\).
AQA M2 2006 June Q7
8 marks Standard +0.3
7 A particle of mass 20 kg moves along a straight horizontal line. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance force of magnitude \(10 \sqrt { v }\) newtons acts on the particle while it is moving. At time \(t = 0\) the velocity of the particle is \(25 \mathrm {~ms} ^ { - 1 }\).
  1. Show that, at time \(t\) $$v = \left( \frac { 20 - t } { 4 } \right) ^ { 2 }$$
  2. State the value of \(t\) when the particle comes to rest.
AQA M2 2007 June Q1
10 marks Moderate -0.8
1 A hot air balloon moves vertically upwards with a constant velocity. When the balloon is at a height of 30 metres above ground level, a box of mass 5 kg is released from the balloon. After the box is released, it initially moves vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\).
  1. Find the initial kinetic energy of the box.
  2. Show that the kinetic energy of the box when it hits the ground is 1720 J .
  3. Hence find the speed of the box when it hits the ground.
  4. State two modelling assumptions which you have made.
AQA M2 2007 June Q2
9 marks Standard +0.3
2 A uniform lamina is in the shape of a rectangle \(A B C D\) and a square \(E F G H\), as shown in the diagram. The length \(A B\) is 20 cm , the length \(B C\) is 30 cm , the length \(D E\) is 5 cm and the length \(E F\) is 10 cm . The point \(P\) is the midpoint of \(A B\) and the point \(Q\) is the midpoint of \(H G\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-2_615_1221_1585_429}
  1. Explain why the centre of mass of the lamina lies on \(P Q\).
  2. Find the distance of the centre of mass of the lamina from \(A B\).
  3. The lamina is freely suspended from \(A\). Find, to the nearest degree, the angle between \(A D\) and the vertical when the lamina is in equilibrium.
AQA M2 2007 June Q3
11 marks Moderate -0.3
3 A particle has mass 800 kg . A single force of \(( 2400 \mathbf { i } - 4800 t \mathbf { j } )\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.
  1. Find the acceleration of the particle at time \(t\).
  2. At time \(t = 0\), the velocity of the particle is \(( 6 \mathbf { i } + 30 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The velocity of the particle at time \(t\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Show that $$\mathbf { v } = ( 6 + 3 t ) \mathbf { i } + \left( 30 - 3 t ^ { 2 } \right) \mathbf { j }$$
  3. Initially, the particle is at the point with position vector \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
AQA M2 2007 June Q4
9 marks Standard +0.3
4 A uniform plank is 10 m long and has mass 15 kg . It is placed on horizontal ground at the edge of a vertical river bank, so that 2 m of the plank is projecting over the edge, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-3_250_1285_1361_388}
  1. A woman of mass 50 kg stands on the part of the plank which projects over the river. Find the greatest distance from the river bank at which she can safely stand.
  2. The woman wishes to stand safely at the end of the plank which projects over the river. Find the minimum mass which she should place on the other end of the plank so that she can do this.
  3. State how you have used the fact that the plank is uniform in your solution.
  4. State one other modelling assumption which you have made.
AQA M2 2007 June Q5
9 marks Standard +0.3
5 A bead of mass \(m\) moves on a smooth circular ring of radius \(a\) which is fixed in a vertical plane, as shown in the diagram. Its speed at \(A\), the highest point of its path, is \(v\) and its speed at \(B\), the lowest point of its path, is \(7 v\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-4_419_317_484_842}
  1. Show that \(v = \sqrt { \frac { a g } { 12 } }\).
  2. Find the reaction of the ring on the bead, in terms of \(m\) and \(g\), when the bead is at \(A\).
AQA M2 2007 June Q6
12 marks Standard +0.3
6 An elastic string has one end attached to a point \(O\), fixed on a horizontal table. The other end of the string is attached to a particle of mass 5 kilograms. The elastic string has natural length 2 metres and modulus of elasticity 200 newtons. The particle is pulled so that it is 2.5 metres from the point \(O\) and it is then released from rest on the table.
  1. Calculate the elastic potential energy when the particle is 2.5 m from the point \(O\).
  2. If the table is smooth, show that the speed of the particle when the string becomes slack is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The table is, in fact, rough and the coefficient of friction between the particle and the table is 0.4 . Find the speed of the particle when the string becomes slack.
AQA M2 2007 June Q7
6 marks Moderate -0.5
7 A stone of mass \(m\) is moving along the smooth horizontal floor of a tank which is filled with a viscous liquid. At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a resistance force of magnitude \(\lambda m v\), where \(\lambda\) is a constant.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v$$
  2. The initial speed of the stone is \(U\). Show that $$v = U \mathrm { e } ^ { - \lambda t }$$
AQA M2 2007 June Q8
9 marks Standard +0.3
8 A particle, \(P\), of mass 3 kg is attached to one end of a light inextensible string. The string passes through a smooth fixed ring, \(O\), and a second particle, \(Q\), of mass 5 kg is attached to the other end of the string. The particle \(Q\) hangs at rest vertically below the ring and the particle \(P\) moves with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, as shown in the diagram. The angle between \(O P\) and the vertical is \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-5_474_476_1425_774}
  1. Explain why the tension in the string is 49 N .
  2. Find \(\theta\).
  3. Find the radius of the horizontal circle.
AQA M2 2010 June Q1
3 marks Easy -1.2
1 A particle moves along a straight line through the origin. At time \(t\), the displacement, \(s\), of the particle from the origin is given by $$s = 5 t ^ { 2 } + 3 \cos 4 t$$ Find the velocity of the particle at time \(t\).
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-03_2484_1709_223_153}
AQA M2 2010 June Q2
9 marks Moderate -0.8
2 John is at the top of a cliff, looking out over the sea. He throws a rock, of mass 3 kg , horizontally with a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The rock falls a vertical distance of 51 metres to reach the surface of the sea.
  1. Calculate the kinetic energy of the rock when it is thrown.
  2. Calculate the potential energy lost by the rock when it reaches the surface of the sea.
    1. Find the kinetic energy of the rock when it reaches the surface of the sea.
    2. Hence find the speed of the rock when it reaches the surface of the sea.
  4. State one modelling assumption which has been made.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-05_2484_1709_223_153}
AQA M2 2010 June Q3
4 marks Moderate -0.3
3 A uniform circular lamina, of radius 4 cm and mass 0.4 kg , has a centre \(O\), and \(A B\) is a diameter. To create a medal, a smaller uniform circular lamina, of radius 2 cm and mass 0.1 kg , is attached so that the centre of the smaller lamina is at the point \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-06_671_878_513_598}
  1. Explain why the centre of mass of the medal is on the line \(A B\).
  2. Find the distance of the centre of mass of the medal from the point \(B\). \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-06_1259_1705_1448_155}
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-07_2484_1709_223_153}
AQA M2 2010 June Q4
12 marks Standard +0.3
4 A particle has mass 200 kg and moves on a smooth horizontal plane. A single horizontal force, \(\left( 400 \cos \left( \frac { \pi } { 2 } t \right) \mathbf { i } + 600 t ^ { 2 } \mathbf { j } \right)\) newtons, acts on the particle at time \(t\) seconds. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find the acceleration of the particle at time \(t\).
  2. When \(t = 4\), the velocity of the particle is \(( - 3 \mathbf { i } + 56 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  3. Find \(t\) when the particle is moving due west.
  4. Find the speed of the particle when it is moving due west.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-09_2484_1709_223_153}
AQA M2 2010 June Q5
7 marks Standard +0.8
5 A particle is moving along a straight line. At time \(t\), the velocity of the particle is \(v\). The acceleration of the particle throughout the motion is \(- \frac { \lambda } { v ^ { \frac { 1 } { 4 } } }\), where \(\lambda\) is a positive constant. The velocity of the particle is \(u\) when \(t = 0\). Find \(v\) in terms of \(u , \lambda\) and \(t\).
(7 marks)
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-10_2078_1719_632_150}
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-11_2484_1709_223_153}
AQA M2 2010 June Q6
13 marks Standard +0.3
6 When a car, of mass 1200 kg , travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons. The car has a maximum constant speed of \(48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road.
  1. Show that the maximum power of the car is 69120 watts.
  2. The car is travelling along a straight horizontal road. Find the maximum possible acceleration of the car when it is travelling at a speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The car starts to descend a hill on a straight road which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal. Find the maximum possible constant speed of the car as it travels on this road down the hill. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-13_2484_1709_223_153} \(7 \quad\) A uniform rod \(A B\), of length 4 m and mass 6 kg , rests in equilibrium with one end, \(A\), on smooth horizontal ground. The rod rests on a rough horizontal peg at the point \(C\), where \(A C\) is 3 m . The rod is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-14_422_984_447_529}
AQA M2 2010 June Q8
7 marks Standard +0.3
8 A particle is attached to one end of a light inextensible string of length 3 metres. The other end of the string is attached to a fixed point \(O\). The particle is set into motion horizontally at point \(P\) with speed \(v\), so that it describes part of a vertical circle whose centre is \(O\). The point \(P\) is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-16_510_334_493_861} The particle first comes momentarily to rest at the point \(Q\), where \(O Q\) makes an angle of \(15 ^ { \circ }\) to the vertical.
  1. Find the value of \(v\).
  2. When the particle is at rest at the point \(Q\), the tension in the string is 22 newtons. Find the mass of the particle.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-17_2484_1709_223_153}
AQA M2 2010 June Q9
8 marks Standard +0.3
9 A particle, of mass 8 kg , is attached to one end of a length of elastic string. The particle is placed on a smooth horizontal surface. The other end of the elastic string is attached to a point \(O\) fixed on the horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 192 N . \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-18_165_789_571_630} The particle is set in motion on the horizontal surface so that it moves in a circle, centre \(O\), with constant speed \(3 \mathrm {~ms} ^ { - 1 }\). Find the radius of the circle. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-19_2349_1691_221_153} \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-20_2505_1730_212_139}
AQA M2 2011 June Q1
7 marks Moderate -0.8
1 In an Olympic diving competition, Kim, who has mass 58 kg , dives from a fixed platform, 10 metres above the surface of the pool. She leaves the platform with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Assume that Kim's weight is the only force that acts on her after she leaves the platform. Kim is to be modelled as a particle which is initially 1 metre above the platform.
  1. Calculate Kim's initial kinetic energy.
  2. By using conservation of energy, find Kim's speed when she is 6 metres below the platform.
AQA M2 2011 June Q2
5 marks Easy -1.3
2 The diagram shows four particles, \(A , B , C\) and \(D\), which are fixed in a horizontal plane which contains the \(x\) - and \(y\)-axes, as shown. Particle \(A\) has mass 2 kg and is attached at the point ( 9,6 ).
Particle \(B\) has mass 3 kg and is attached at the point ( 2,4 ).
Particle \(C\) has mass 8 kg and is attached at the point \(( 3,8 )\).
Particle \(D\) has mass 7 kg and is attached at the point \(( 6,11 )\). \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-2_748_774_1402_625} Find the coordinates of the centre of mass of the four particles.
AQA M2 2011 June Q3
14 marks Standard +0.3
3 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \mathrm { e } ^ { - 2 t } \mathbf { i } + \left( 6 t - 3 t ^ { 2 } \right) \mathbf { j }$$
  1. Find an expression for the acceleration of the particle at time \(t\).
  2. The mass of the particle is 5 kg .
    1. Find an expression for the force \(\mathbf { F }\) acting on the particle at time \(t\).
    2. Find the magnitude of \(\mathbf { F }\) when \(t = 0\).
  3. Find the value of \(t\) when \(\mathbf { F }\) acts due west.
  4. When \(t = 0\), the particle is at the point with position vector \(( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
AQA M2 2011 June Q4
7 marks Moderate -0.3
4 Ken is trying to cross a river of width 4 m . He has a uniform plank, \(A B\), of length 8 m and mass 17 kg . The ground on both edges of the river bank is horizontal. The plank rests at two points, \(C\) and \(D\), on fixed supports which are on opposite sides of the river. The plank is at right angles to both river banks and is horizontal. The distance \(A C\) is 1 m , and the point \(C\) is at a horizontal distance of 0.6 m from the river bank. Ken, who has mass 65 kg , stands on the plank directly above the middle of the river, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-3_468_1086_1710_479}
  1. Draw a diagram to show the forces acting on the plank.
  2. Given that the reaction on the plank at the point \(D\) is \(44 g \mathrm {~N}\), find the horizontal distance of the point \(D\) from the nearest river bank.
  3. State how you have used the fact that the plank is uniform in your solution.
AQA M2 2011 June Q5
4 marks Easy -1.2
5 A train consists of an engine and five carriages. A constant resistance force of 3000 N acts on the engine, and a constant resistance force of 400 N acts on each of the five carriages. The maximum speed of the train on a horizontal track is \(90 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
  1. Show that this speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Hence find the maximum power output of the engine. Give your answer in kilowatts.
    (3 marks)
AQA M2 2011 June Q6
6 marks Standard +0.3
6 A car, of mass \(m \mathrm {~kg}\), is moving along a straight horizontal road. At time \(t\) seconds, the car has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the car moves, it experiences a resistance force of magnitude \(2 m v ^ { \frac { 5 } { 4 } }\) newtons. No other horizontal force acts on the car.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 2 v ^ { \frac { 5 } { 4 } }$$ (1 mark)
  2. The initial speed of the car is \(16 \mathrm {~ms} ^ { - 1 }\). Show that $$v = \left( \frac { 2 } { t + 1 } \right) ^ { 4 }$$ (5 marks)
AQA M2 2011 June Q7
8 marks Standard +0.3
7 Two light inextensible strings each have one end attached to a particle, \(P\), of mass 4 kg . The other ends of the strings are attached to the fixed points \(A\) and \(B\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed in a horizontal circle. The centre, \(C\), of this circle is directly below the point \(B\). The two strings are inclined at \(30 ^ { \circ }\) and \(50 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut. As the particle moves in the horizontal circle, the tension in the string \(B P\) is 20 N . \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-5_750_469_742_781}
  1. Find the tension in the string \(A P\).
  2. The speed of the particle is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the length of \(C P\), the radius of the horizontal circle.