Questions M2 (1537 questions)

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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 \(4 \mathrm {~ms} ^ { - 1 }\).
Edexcel M2 2024 October Q1
Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) is moving with velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = 3 ( t + 2 ) ^ { 2 } \mathbf { i } + 5 t ( t + 2 ) \mathbf { j }$$ Position vectors are given relative to the fixed point \(O\) At time \(t = 0 , P\) is at the point with position vector \(( - 30 \mathbf { i } - 45 \mathbf { j } ) \mathrm { m }\).
  1. Find the position vector of \(P\) when \(t = 3\)
  2. Find the magnitude of the acceleration of \(P\) when \(t = 3\) At time \(T\) seconds, \(P\) is moving in the direction of the vector \(2 \mathbf { i } + \mathbf { j }\)
  3. Find the value of \(T\)
Edexcel M2 2024 October Q2
Standard +0.3
  1. A particle \(Q\) of mass 3 kg is moving on a smooth horizontal surface.
Particle \(Q\) is moving with velocity \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives a horizontal impulse of magnitude \(3 \sqrt { 82 } \mathrm { Ns }\). Immediately after receiving the impulse, the velocity of \(Q\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(x\) and \(y\) are positive constants. The kinetic energy gained by \(Q\) as a result of receiving the impulse is 138 J .
Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(Q\) immediately after receiving the impulse.
Edexcel M2 2024 October Q3
Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-06_275_1143_303_461} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A van of mass 900 kg is moving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 25 }\). The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 240 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 15 kW .
  1. Find the acceleration of the van at the instant when the speed of the van is \(12 \mathrm {~ms} ^ { - 1 }\) At the instant when the speed of the van is \(14 \mathrm {~ms} ^ { - 1 }\), the trailer is passing the point \(A\) on the slope and the towbar breaks. The trailer continues to move up the slope until it comes to rest at the point \(B\).
    The resistance to the motion of the trailer from non-gravitational forces is still modelled as a constant force of magnitude 240 N .
  2. Use the work-energy principle to find the distance \(A B\).
Edexcel M2 2024 October Q4
Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-10_301_871_319_598} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(A B C D\) shown in Figure 2 is in the shape of an isosceles trapezium.
  • \(B C\) is parallel to \(A D\) and angle \(B A D\) is equal to angle \(A D C\)
  • \(B C = 5 a\) and \(A D = 7 a\)
  • the perpendicular distance between \(B C\) and \(A D\) is \(3 a\)
  • the distance of the centre of mass of \(A B C D\) from \(A D\) is \(d\)
    1. Show that \(d = \frac { 17 } { 12 } a\)
The uniform lamina \(P Q R S\) is a rectangle with \(P Q = 5 a\) and \(Q R = 9 a\).
The lamina \(A B C D\) in Figure 2 is used to cut a hole in \(P Q R S\) to form the template shown shaded in Figure 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-10_364_876_1567_593} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
The template is freely suspended from \(P\) and hangs in equilibrium with \(P S\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.
  • Find the value of \(\theta\)
    •
    Edexcel M2 2024 October Q5
    Standard +0.3
    1. The fixed points \(X\) and \(Y\) lie on horizontal ground.
    At time \(t = 0\), a particle \(P\) is projected from \(X\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at angle \(\theta\) to the ground. Particle \(P\) moves freely under gravity and first hits the ground at \(Y\).
    1. Show that \(X Y = \frac { u ^ { 2 } \sin 2 \theta } { g }\) The points \(A\) and \(B\) lie on horizontal ground. A vertical pole \(C D\) has length 5 m .
      The end \(C\) is fixed to the ground between \(A\) and \(B\), where \(A C = 12 \mathrm {~m}\).
      At time \(t = 0\), a particle \(Q\) is projected from \(A\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to the ground.
      Particle \(Q\) moves freely under gravity, passes over the pole and first hits the ground at \(B\), as shown in Figure 4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-14_335_1179_1032_443} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure}
    2. Find the distance \(C B\).
    3. Find the height of \(Q\) above \(D\) at the instant when \(Q\) passes over the pole.
    Edexcel M2 2024 October Q6
    Standard +0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-18_419_1307_315_379} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} A uniform beam \(A B\), of weight \(5 W\) and length \(12 a\), rests with end \(A\) on rough horizontal ground.
    A package of weight \(W\) is attached to the beam at \(B\).
    The beam rests in equilibrium on a smooth horizontal peg at \(C\), with \(A C = 9 a\), as shown in Figure 5.
    The beam is inclined at an angle \(\theta\) to the ground, where \(\tan \theta = \frac { 5 } { 12 }\) The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg. The package is modelled as a particle. The normal reaction between the beam and the peg at \(C\) has magnitude \(k W\) Using the model,
    1. show that \(k = \frac { 56 } { 13 }\) The coefficient of friction between \(A\) and the ground is \(\mu\) Given that the beam is resting in limiting equilibrium,
    2. find the value of \(\mu\)
    Edexcel M2 2024 October Q7
    Standard +0.3
    1. A particle \(P\) has mass \(5 m\) and a particle \(Q\) has mass \(2 m\).
    The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.
    Particle \(P\) collides directly with particle \(Q\).
    Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
    The direction of motion of \(Q\) is reversed as a result of the collision.
    The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    1. Find the set of values of \(e\) for which the direction of motion of \(P\) is unchanged as a result of the collision. In the collision, \(Q\) receives an impulse of magnitude \(\frac { 60 } { 7 } m u\)
    2. Show that \(e = \frac { 1 } { 5 }\) After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds and there is a second collision between \(P\) and \(Q\).
      The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
    3. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the second collision between \(P\) and \(Q\).
    CAIE M2 2010 June Q1
    4 marks Moderate -0.3
    \includegraphics{figure_1} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg, and a uniform straight wire of length 40 cm and mass 0.9 kg. The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire. [4]
    CAIE M2 2010 June Q2
    5 marks Standard +0.3
    \includegraphics{figure_2} A uniform solid cone has height 30 cm and base radius \(r\) cm. The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35°\), when the cone topples. The diagram shows a cross-section of the cone.
    1. Find the value of \(r\). [3]
    2. Show that the coefficient of friction between the cone and the plane is greater than 0.7. [2]
    CAIE M2 2010 June Q3
    6 marks Standard +0.3
    \includegraphics{figure_3} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T\) N. The acceleration of the particle has magnitude \(7.5 \text{ m s}^{-2}\).
    1. Show that \(\tan \theta = 0.75\) and find the value of \(T\). [4]
    2. Find the speed of the particle. [2]
    CAIE M2 2010 June Q4
    5 marks Standard +0.3
    \includegraphics{figure_4} A uniform lamina of weight 15 N is in the form of a trapezium \(ABCD\) with dimensions as shown in the diagram. The lamina is freely hinged at \(A\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(B\). The lamina is in equilibrium with \(AB\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30°\) upwards from the horizontal (see diagram). Find the tension in the string. [5]
    CAIE M2 2010 June Q5
    9 marks Standard +0.3
    A particle is projected from a point \(O\) on horizontal ground. The velocity of projection has magnitude \(20 \text{ m s}^{-1}\) and direction upwards at an angle \(\theta\) to the horizontal. The particle passes through the point which is 7 m above the ground and 16 m horizontally from \(O\), and hits the ground at the point \(A\).
    1. Using the equation of the particle's trajectory and the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), show that the possible values of \(\tan \theta\) are \(\frac{4}{3}\) and \(\frac{1}{4}\). [4]
    2. Find the distance \(OA\) for each of the two possible values of \(\tan \theta\). [3]
    3. Sketch in the same diagram the two possible trajectories. [2]
    CAIE M2 2010 June Q6
    10 marks Standard +0.3
    \includegraphics{figure_6} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m. The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(AB\) (see diagram).
    1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N. [4]
    \(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.
    1. Find the speed with which \(P\) passes through \(M\). [6]