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AQA M2 2008 June Q5
12 marks Moderate -0.3
5 A particle moves on a horizontal plane in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the particle's position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = 8 \left( \cos \frac { 1 } { 4 } t \right) \mathbf { i } - 8 \left( \sin \frac { 1 } { 4 } t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
  2. Show that the speed of the particle is a constant.
  3. Prove that the particle is moving in a circle.
  4. Find the angular speed of the particle.
  5. Find an expression for the acceleration of the particle at time \(t\).
  6. State the magnitude of the acceleration of the particle.
AQA M2 2008 June Q6
8 marks Moderate -0.5
6 A car, of mass \(m\), is moving along a straight smooth horizontal road. At time \(t\), the car has speed \(v\). As the car moves, it experiences a resistance force of magnitude \(0.05 m v\). No other horizontal force acts on the car.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.05 v$$
  2. When \(t = 0\), the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(v = 20 \mathrm { e } ^ { - 0.05 t }\).
  3. Find the time taken for the speed of the car to reduce to \(10 \mathrm {~ms} ^ { - 1 }\).
AQA M2 2008 June Q7
9 marks Standard +0.3
7 A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string, of length \(a\). The bead is then set into circular motion with the string taut at \(B\), where \(B\) is vertically below \(O\), with a horizontal speed \(u\). \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-5_451_458_461_760}
  1. Given that the string does not become slack, show that the least value of \(u\) required for the bead to make complete revolutions about \(O\) is \(\sqrt { 5 a g }\).
  2. In the case where \(u = \sqrt { 5 a g }\), find, in terms of \(g\) and \(m\), the tension in the string when the bead is at the point \(C\), which is at the same horizontal level as \(O\), as shown in the diagram.
  3. State one modelling assumption that you have made in your solution.
AQA M2 2008 June Q8
16 marks Standard +0.3
8
  1. Hooke's law states that the tension in a stretched string of natural length \(l\) and modulus of elasticity \(\lambda\) is \(\frac { \lambda x } { l }\) when its extension is \(x\). Using this formula, prove that the work done in stretching a string from an unstretched position to a position in which its extension is \(e\) is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
    (3 marks)
  2. A particle, of mass 5 kg , is attached to one end of a light elastic string of natural length 0.6 metres and modulus of elasticity 150 N . The other end of the string is fixed to a point \(O\).
    1. Find the extension of the elastic string when the particle hangs in equilibrium directly below \(O\).
    2. The particle is pulled down and held at the point \(P\), which is 0.9 metres vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 11.25 J .
    3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres above \(\boldsymbol { P }\). Show that, while the string is taut, $$v ^ { 2 } = 10.4 x - 50 x ^ { 2 }$$
    4. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.
AQA M2 2009 June Q1
9 marks Moderate -0.5
1 A particle moves under the action of a force, \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of the particle is given by $$\mathbf { v } = \left( t ^ { 3 } - 15 t - 5 \right) \mathbf { i } + \left( 6 t - t ^ { 2 } \right) \mathbf { j }$$
  1. Find an expression for the acceleration of the particle at time \(t\).
  2. The mass of the particle is 4 kg .
    1. Show that, at time \(t\), $$\mathbf { F } = \left( 12 t ^ { 2 } - 60 \right) \mathbf { i } + ( 24 - 8 t ) \mathbf { j }$$
    2. Find the magnitude of \(\mathbf { F }\) when \(t = 2\).
AQA M2 2009 June Q2
9 marks Moderate -0.8
2 A slide at a water park may be modelled as a smooth plane of length 20 metres inclined at \(30 ^ { \circ }\) to the vertical. Anne, who has a mass of 55 kg , slides down the slide. At the top of the slide, she has an initial velocity of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the slide.
  1. Calculate Anne's initial kinetic energy.
  2. By using conservation of energy, find the kinetic energy and the speed of Anne after she has travelled the 20 metres.
  3. State one modelling assumption which you have made.
AQA M2 2009 June Q3
9 marks Standard +0.3
3 A uniform ladder, of length 6 metres and mass 22 kg , rests with its foot, \(A\), on a rough horizontal floor and its top, \(B\), leaning against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the floor is \(\theta\). A man, of mass 90 kg , is standing at point \(C\) on the ladder so that the distance \(A C\) is 5 metres. With the man in this position, the ladder is on the point of slipping. The coefficient of friction between the ladder and the horizontal floor is 0.6 . The man may be modelled as a particle at \(C\). \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-3_707_702_742_646}
  1. Show that the magnitude of the frictional force between the ladder and the horizontal floor is 659 N , correct to three significant figures.
  2. Find the angle \(\theta\).
AQA M2 2009 June Q4
8 marks Standard +0.3
4 Two light inextensible strings each have one end attached to a particle, \(P\), of mass 6 kg . The other ends of the strings are attached to the fixed points \(B\) and \(C\). The point \(C\) is vertically above the point \(B\). The particle moves, at constant speed, in a horizontal circle, with centre 0.6 m below point \(B\), with the strings inclined at \(40 ^ { \circ }\) and \(60 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-4_761_542_539_751}
  1. As the particle moves in the horizontal circle, the tensions in the two strings are equal. Show that the tension in the strings is 46.4 N , correct to three significant figures.
  2. Find the speed of the particle.
AQA M2 2009 June Q5
6 marks Moderate -0.8
5 A train, of mass 600 tonnes, travels at constant speed up a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 40 }\). The speed of the train is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it experiences total resistance forces of 200000 N . Find the power produced by the train, giving your answer in kilowatts.
AQA M2 2009 June Q6
12 marks Standard +0.3
6 A block, of mass 5 kg , is attached to one end of a length of elastic string. The other end of the string is fixed to a vertical wall. The block is placed on a horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 180 N . The block is pulled so that it is 2 m from the wall and is then released from rest. Whilst taut, the string remains horizontal. It may be assumed that, after the string becomes slack, it does not interfere with the movement of the block. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-5_396_960_660_534}
  1. Calculate the elastic potential energy when the block is 2 m from the wall.
  2. If the horizontal surface is smooth, find the speed of the block when it hits the wall.
  3. The surface is in fact rough and the coefficient of friction between the block and the surface is \(\mu\). Find \(\mu\) if the block comes to rest just as it reaches the wall.
AQA M2 2009 June Q7
10 marks Standard +0.3
7 In crazy golf, a golf ball is hit so that it starts to move in a vertical circle on the inside of a smooth cylinder. Model the golf ball as a particle, \(P\), of mass \(m\). The circular path of the golf ball has radius \(a\) and centre \(O\). At time \(t\), the angle between \(O P\) and the horizontal is \(\theta\), as shown in the diagram. The golf ball has speed \(u\) at the lowest point of its circular path. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-6_739_742_719_641}
  1. Show that, while the golf ball is in contact with the cylinder, the reaction of the cylinder on the golf ball is $$\frac { m u ^ { 2 } } { a } - 3 m g \sin \theta - 2 m g$$
  2. Given that \(u = \sqrt { 3 a g }\), the golf ball will not complete a vertical circle inside the cylinder. Find the angle which \(O P\) makes with the horizontal when the golf ball leaves the surface of the cylinder.
    (4 marks)
AQA M2 2009 June Q8
12 marks Standard +0.3
8 A stone, of mass \(m\), is moving in a straight line along smooth horizontal ground.
At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a total resistance force of magnitude \(\lambda m v ^ { \frac { 3 } { 2 } }\), where \(\lambda\) is a constant. No other horizontal force acts on the stone.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v ^ { \frac { 3 } { 2 } }$$ (2 marks)
  2. The initial speed of the stone is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that $$v = \frac { 36 } { ( 2 + 3 \lambda t ) ^ { 2 } }$$ (7 marks)
  3. Find, in terms of \(\lambda\), the time taken for the speed of the stone to drop to \(4 \mathrm {~ms} ^ { - 1 }\).
AQA M3 2009 June Q1
5 marks Standard +0.3
1 A ball of mass \(m\) is travelling vertically downwards with speed \(u\) when it hits a horizontal floor. The ball bounces vertically upwards to a height \(h\). It is thought that \(h\) depends on \(m , u\), the acceleration due to gravity \(g\), and a dimensionless constant \(k\), such that $$h = k m ^ { \alpha } u ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, find the values of \(\alpha , \beta\) and \(\gamma\).
AQA M3 2009 June Q2
10 marks Standard +0.3
2 A particle is projected from a point \(O\) on a horizontal plane and has initial velocity components of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(10 \mathrm {~ms} ^ { - 1 }\) parallel to and perpendicular to the plane respectively. At time \(t\) seconds after projection, the horizontal and upward vertical distances of the particle from the point \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = - \frac { g } { 8 } x ^ { 2 } + 5 x$$
  2. By using the equation in part (a), find the horizontal distance travelled by the particle whilst it is more than 1 metre above the plane.
  3. Hence find the time for which the particle is more than 1 metre above the plane.
AQA M3 2009 June Q3
14 marks Standard +0.8
3 A fishing boat is travelling between two ports, \(A\) and \(B\), on the shore of a lake. The bearing of \(B\) from \(A\) is \(130 ^ { \circ }\). The fishing boat leaves \(A\) and travels directly towards \(B\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A patrol boat on the lake is travelling with speed \(4 \mathrm {~ms} ^ { - 1 }\) on a bearing of \(040 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-3_713_1319_443_406}
  1. Find the velocity of the fishing boat relative to the patrol boat, giving your answer as a speed together with a bearing.
  2. When the patrol boat is 1500 m due west of the fishing boat, it changes direction in order to intercept the fishing boat in the shortest possible time.
    1. Find the bearing on which the patrol boat should travel in order to intercept the fishing boat.
    2. Given that the patrol boat intercepts the fishing boat before it reaches \(B\), find the time, in seconds, that it takes the patrol boat to intercept the fishing boat after changing direction.
    3. State a modelling assumption necessary for answering this question, other than the boats being particles.
AQA M3 2009 June Q4
10 marks Standard +0.3
4 A particle of mass 0.5 kg is initially at rest. The particle then moves in a straight line under the action of a single force. This force acts in a constant direction and has magnitude \(\left( t ^ { 3 } + t \right) \mathrm { N }\), where \(t\) is the time, in seconds, for which the force has been acting.
  1. Find the magnitude of the impulse exerted by the force on the particle between the times \(t = 0\) and \(t = 4\).
  2. Hence find the speed of the particle when \(t = 4\).
  3. Find the time taken for the particle to reach a speed of \(12 \mathrm {~ms} ^ { - 1 }\).
AQA M3 2009 June Q5
12 marks Challenging +1.2
5 Two smooth spheres, \(A\) and \(B\), of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres of the spheres, and \(B\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_314_1100_593_392} \captionsetup{labelformat=empty} \caption{Before collision}
\end{figure} Immediately after the collision, \(A\) and \(B\) move with speeds \(u\) and \(v\) in directions which make angles of \(90 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the line of centres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}
  1. Show that \(v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  2. Find the coefficient of restitution between the spheres.
  3. Given that the mass of \(A\) is 0.5 kg , show that the magnitude of the impulse exerted on \(A\) during the collision is 2.17 Ns , correct to three significant figures.
  4. Find the mass of \(B\).
AQA M3 2009 June Q6
13 marks Standard +0.3
6 A smooth sphere \(A\) of mass \(m\) is moving with speed \(5 u\) in a straight line on a smooth horizontal table. The sphere \(A\) collides directly with a smooth sphere \(B\) of mass \(7 m\), having the same radius as \(A\) and moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-5_287_880_529_571}
  1. Show that the speed of \(B\) after the collision is \(\frac { u } { 2 } ( e + 3 )\).
  2. Given that the direction of motion of \(A\) is reversed by the collision, show that \(e > \frac { 3 } { 7 }\).
  3. Subsequently, \(B\) hits a wall fixed at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). Given that after \(B\) rebounds from the wall both spheres move in the same direction and collide again, show also that \(e < \frac { 9 } { 13 }\).
    (4 marks)
AQA M3 2009 June Q7
11 marks Challenging +1.2
7 A particle is projected from a point \(O\) on a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The particle is projected down the plane with velocity \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the plane and first strikes it at a point \(A\). The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-6_449_963_488_529}
  1. Show that the time taken by the particle to travel from \(O\) to \(A\) is $$\frac { 20 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
  2. Find the components of the velocity of the particle parallel to and perpendicular to the slope as it hits the slope at \(A\).
  3. The coefficient of restitution between the slope and the particle is 0.5 . Find the speed of the particle as it rebounds from the slope.
Edexcel M4 Q1
13 marks Challenging +1.2
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-03_457_638_233_598} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A fixed smooth plane is inclined to the horizontal at an angle of \(45 ^ { \circ }\). A particle \(P\) is moving horizontally and strikes the plane. Immediately before the impact, \(P\) is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, \(P\) is moving in a direction which makes an angle of \(30 ^ { \circ }\) with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of \(P\) which is lost in the impact.
Edexcel M4 Q2
8 marks Challenging +1.8
2. At time \(t = 0\), a particle \(P\) of mass \(m\) is projected vertically upwards with speed \(\sqrt { \frac { g } { k } }\), where \(k\) is a constant. At time \(t\) the speed of \(P\) is \(v\). The particle \(P\) moves against air resistance whose magnitude is modelled as being \(m k v ^ { 2 }\) when the speed of \(P\) is \(v\). Find, in terms of \(k\), the distance travelled by \(P\) until its speed first becomes half of its initial speed.
Edexcel M4 Q3
10 marks Challenging +1.2
At noon a motorboat \(P\) is 2 km north-west of another motorboat \(Q\). The motorboat \(P\) is moving due south at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motorboat \(Q\) is pursuing motorboat \(P\) at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and sets a course in order to get as close to motorboat \(P\) as possible.
  1. Find the course set by \(Q\), giving your answer as a bearing to the nearest degree.
  2. Find the shortest distance between \(P\) and \(Q\).
  3. Find the distance travelled by \(Q\) from its position at noon to the point of closest approach.
Edexcel M4 Q4
10 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-08_479_807_246_571} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.
  1. Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
  2. Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
  3. Determine the stability of the position of equilibrium.
Edexcel M4 Q5
8 marks Standard +0.8
Two small smooth spheres \(A\) and \(B\), of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Show that the velocity of \(B\) immediately after the collision is \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the impulse of \(B\) on \(A\) in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to \(\mathbf { i } + \mathbf { j }\).
  3. Find the coefficient of restitution between \(A\) and \(B\).
Edexcel M4 Q6
14 marks Challenging +1.2
6. A light elastic spring \(A B\) has natural length \(2 a\) and modulus of elasticity \(2 m n ^ { 2 } a\), where \(n\) is a constant. A particle \(P\) of mass \(m\) is attached to the end \(A\) of the spring. At time \(t = 0\), the spring, with \(P\) attached, lies at rest and unstretched on a smooth horizontal plane. The other end \(B\) of the spring is then pulled along the plane in the direction \(A B\) with constant acceleration \(f\). At time \(t\) the extension of the spring is \(x\).
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
  2. Find \(x\) in terms of \(n , f\) and \(t\). Hence find
  3. the maximum extension of the spring,
  4. the speed of \(P\) when the spring first reaches its maximum extension.
    1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively]
    A man cycles at a constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on level ground and finds that when his velocity is \(u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the velocity of the wind appears to be \(v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(v\) is a positive constant. When the man cycles with velocity \(\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), the velocity of the wind appears to be \(w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(w\) is a positive constant. Find, in terms of \(u\), the true velocity of the wind.