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AQA M2 2016 June Q1
8 marks Moderate -0.8
A stone, of mass \(0.3\) kg, is thrown with a speed of \(8 \text{ m s}^{-1}\) from a point at a height of \(5\) metres above a horizontal surface.
  1. Calculate the initial kinetic energy of the stone. [2 marks]
    1. Find the kinetic energy of the stone when it hits the surface. [3 marks]
    2. Hence find the speed of the stone when it hits the surface. [2 marks]
    3. State one modelling assumption that you have made. [1 mark]
AQA M2 2016 June Q2
13 marks Standard +0.3
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, \(\mathbf{v} \text{ m s}^{-1}\), is given by $$\mathbf{v} = (8t - t^4)\mathbf{i} + 6e^{-3t}\mathbf{j}$$
  1. Find an expression for the acceleration of the particle at time \(t\). [2 marks]
  2. The mass of the particle is \(2\) kg.
    1. Find an expression for the force \(\mathbf{F}\) acting on the particle at time \(t\). [2 marks]
    2. Find the magnitude of \(\mathbf{F}\) when \(t = 1\). [3 marks]
  3. Find the value of \(t\) when \(\mathbf{F}\) acts due south. [2 marks]
  4. When \(t = 0\), the particle is at the point with position vector \((3\mathbf{i} - 5\mathbf{j})\) metres. Find an expression for the position vector, \(\mathbf{r}\) metres, of the particle at time \(t\). [4 marks]
AQA M2 2016 June Q3
9 marks Moderate -0.3
The diagram shows a uniform lamina \(ABCDEFGHIJKL\). \includegraphics{figure_3}
  1. Explain why the centre of mass of the lamina is \(35\) cm from \(AL\). [1 mark]
  2. Find the distance of the centre of mass from \(AF\). [4 marks]
  3. The lamina is freely suspended from \(A\). Find the angle between \(AB\) and the vertical when the lamina is in equilibrium. [3 marks]
  4. Explain, briefly, how you have used the fact that the lamina is uniform. [1 mark]
AQA M2 2016 June Q4
8 marks Standard +0.3
A particle \(P\), of mass \(6\) kg, is attached to one end of a light inextensible string. The string passes through a small smooth ring, fixed at a point \(O\). A second particle \(Q\), of mass \(8\) 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 \(5 \text{ m s}^{-1}\) in a horizontal circle, as shown in the diagram. The angle between \(OP\) and the vertical is \(\theta\). \includegraphics{figure_4}
  1. Find the tension in the string. [1 mark]
  2. Find \(\theta\). [3 marks]
  3. Find the radius of the horizontal circle. [4 marks]
AQA M2 2016 June Q5
12 marks Standard +0.3
A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(R\) vertically below \(O\). The particle is set into motion with a horizontal velocity \(u\) so that it moves in a complete vertical circle with centre \(O\). The point \(T\) on the circle is such that angle \(ROT\) is \(30°\), as shown in the diagram. \includegraphics{figure_5}
  1. Find, in terms of \(g\), \(l\) and \(u\), the speed of the particle at the point \(T\). [3 marks]
  2. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle is at the point \(T\). [3 marks]
  3. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle returns to the point \(R\). [2 marks]
  4. The particle makes complete revolutions. Find, in terms of \(g\) and \(l\), the minimum value of \(u\). [4 marks]
AQA M2 2016 June Q6
8 marks Standard +0.3
A stone, of mass \(m\), falls vertically downwards under gravity through still water. At time \(t\), the stone has speed \(v\) and it experiences a resistance force of magnitude \(\lambda mv\), where \(\lambda\) is a constant.
  1. Show that $$\frac{\text{d}v}{\text{d}t} = g - \lambda v$$ [2 marks]
  2. The initial speed of the stone is \(u\). Find an expression for \(v\) at time \(t\). [6 marks]
AQA M2 2016 June Q7
9 marks Standard +0.8
A uniform ladder, of weight \(W\), rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is \(\mu\) and the coefficient of friction between the ground and the ladder is \(2\mu\). When the ladder is on the point of slipping, the ladder is inclined at an angle of \(\theta\) to the horizontal.
  1. Draw a diagram to show the forces acting on the ladder. [2 marks]
  2. Find \(\tan \theta\) in terms of \(\mu\). [7 marks]
AQA M2 2016 June Q8
8 marks Challenging +1.8
A particle \(P\), of mass \(5\) kg is placed at the point \(A\) on a rough plane which is inclined at \(30°\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(QR = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(AQ = 4\) metres and \(AR = 11\) metres. The three points \(Q\), \(A\) and \(R\) are on a line of greatest slope of the plane. \includegraphics{figure_8} The particle is attached to two light elastic strings, \(PQ\) and \(PR\). One of the strings, \(PQ\), has natural length \(4\) metres and modulus of elasticity \(160\) N, the other string, \(PR\), has natural length \(6\) metres and modulus of elasticity \(120\) N. The particle is released from rest at the point \(A\). The coefficient of friction between the particle and the plane is \(0.4\). Find the distance of the particle from \(Q\) when it is next at rest. [8 marks]
Edexcel M2 Q1
4 marks Moderate -0.8
A constant force acts on a particle of mass 200 grams, moving it 50 cm in a straight line on a rough horizontal surface at a constant speed. The coefficient of friction between the particle and the surface is \(\frac{1}{4}\). Calculate, in J, the work done by the force. [4 marks]
Edexcel M2 Q2
5 marks Moderate -0.3
A stone, of mass 0.9 kg, is projected vertically upwards with speed 10 ms\(^{-1}\) in a medium which exerts a constant resistance to motion. It comes to rest after rising a distance of 3.75 m. Find the magnitude of the non-gravitational resisting force acting on the stone. [5 marks]
Edexcel M2 Q3
8 marks Moderate -0.8
A particle \(P\), of mass 0.4 kg, moves in a straight line such that, at time \(t\) seconds after passing through a fixed point \(O\), its distance from \(O\) is \(x\) metres, where \(x = 3t^2 + 8t\).
  1. Show that \(P\) never returns to \(O\). [2 marks]
  2. Find the value of \(t\) when \(P\) has velocity 20 ms\(^{-1}\). [3 marks]
  3. Show that the force acting on \(P\) is constant, and find its magnitude. [3 marks]
Edexcel M2 Q4
9 marks Standard +0.3
Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(3m\) respectively, are moving on a smooth horizontal table with velocities \((3\mathbf{i} - \mathbf{j})\) ms\(^{-1}\) and \((4\mathbf{i} + \mathbf{j})\) ms\(^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. They collide, after which \(A\) has velocity \((5\mathbf{i} + \mathbf{j})\) ms\(^{-1}\).
  1. Find the magnitude of the impulse exerted on \(B\) by \(A\), stating the units of your answer. [4 marks]
  2. Find the speed of \(B\) immediately after the collision. [5 marks]
Edexcel M2 Q5
10 marks Standard +0.3
A small car, of mass 850 kg, moves on a straight horizontal road. Its engine is working at its maximum rate of 25 kW, and a constant resisting force of magnitude 900 N opposes the car's motion.
  1. Find the acceleration of the car when it is moving with speed 15 ms\(^{-1}\). [3 marks]
  2. Find the maximum speed of the car on the horizontal road. [3 marks]
With the engine still working at 25 kW and the non-gravitational resistance remaining at 900 N, the car now climbs a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{10}\).
  1. Find the maximum speed of the car on this hill. [4 marks]
Edexcel M2 Q6
12 marks Standard +0.3
A uniform wire \(ABCD\) is bent into the shape shown, where the sections \(AB\), \(BC\) and \(CD\) are straight and of length \(3a\), \(10a\) and \(5a\) respectively and \(AD\) is parallel to \(BC\). \includegraphics{figure_6}
  1. Show that the cosine of angle \(BCD\) is \(\frac{3}{5}\). [2 marks]
  2. Find the distances of the centre of mass of the bent wire from (i) \(AB\), (ii) \(BC\). [6 marks]
The wire is hung over a smooth peg at \(B\) and rests in equilibrium.
  1. Find, to the nearest 0.1°, the angle between \(BC\) and the vertical in this position. [4 marks]
Edexcel M2 Q7
12 marks Standard +0.3
Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. \(P\) has speed 4 ms\(^{-1}\). They collide directly. After the collision the direction of motion of both particles has been reversed, and \(Q\) has speed 2 ms\(^{-1}\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). Find
  1. the speed of \(Q\) before the collision, [4 marks]
  2. the speed of \(P\) after the collision, [4 marks]
  3. the kinetic energy, in J, lost in the impact. [4 marks]
Edexcel M2 Q8
15 marks Standard +0.3
In a fairground game, a contestant bowls a ball at a coconut 6 metres away on the same horizontal level. The ball is thrown with an initial speed of 8 ms\(^{-1}\) in a direction making an angle of 30° with the horizontal. \includegraphics{figure_8}
  1. Find the time taken by the ball to travel 6 m horizontally. [2 marks]
  2. Showing your method clearly, decide whether or not the ball will hit the coconut. [4 marks]
  3. Find the greatest height reached by the ball above the level from which it was thrown. [4 marks]
  4. Find the maximum horizontal distance from which it is possible to hit the coconut if the ball is thrown with the same initial speed of 8 m s\(^{-1}\). [3 marks]
  5. State two assumptions that you have made about the ball and the forces which act on it as it travels towards the coconut. [2 marks]
Edexcel M2 Q1
6 marks Moderate -0.3
A ball, of mass \(m\) kg, is moving with velocity \((5\mathbf{i} - 3\mathbf{j})\) ms\(^{-1}\) when it receives an impulse of \((-2\mathbf{i} - 4\mathbf{j})\) Ns. Immediately after the impulse is applied, the ball has velocity \((3\mathbf{i} + k\mathbf{j})\) ms\(^{-1}\). Find the values of the constants \(k\) and \(m\). [6 marks]
Edexcel M2 Q2
6 marks Moderate -0.3
A particle \(P\), initially at rest at the point \(O\), moves in a straight line such that at time \(t\) seconds after leaving \(O\) its acceleration is \((12t - 15)\) ms\(^{-2}\). Find
  1. the velocity of \(P\) at time \(t\) seconds after it leaves \(O\), [3 marks]
  2. the value of \(t\) when the speed of \(P\) is 36 ms\(^{-1}\). [3 marks]
Edexcel M2 Q3
7 marks Standard +0.3
A non-uniform ladder \(AB\), of length \(3a\), has its centre of mass at \(G\), where \(AG = 2a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(AB\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac{14}{9}\). \includegraphics{figure_3} Calculate the coefficient of friction between the ladder and the ground. [7 marks]
Edexcel M2 Q4
9 marks Moderate -0.8
A particle \(P\) starts from the point \(O\) and moves such that its position vector \(\mathbf{r}\) m relative to \(O\) after \(t\) seconds is given by \(\mathbf{r} = at^2\mathbf{i} + bt\mathbf{j}\). 60 seconds after \(P\) leaves \(O\) it is at the point \(Q\) with position vector \((90\mathbf{i} + 30\mathbf{j})\) m.
  1. Find the values of the constants \(a\) and \(b\). [3 marks]
  2. Find the speed of \(P\) when it is at \(Q\). [4 marks]
  3. Sketch the path followed by \(P\) for \(0 \leq t \leq 60\). [2 marks]
Edexcel M2 Q5
10 marks Standard +0.3
A lorry of mass 4200 kg can develop a maximum power of 84 kW. On any road the lorry experiences a non-gravitational resisting force which is directly proportional to its speed. When the lorry is travelling at 20 ms\(^{-1}\) the resisting force has magnitude 2400 N. Find the maximum speed of the lorry when it is
  1. travelling on a horizontal road, [4 marks]
  2. climbing a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{7}\). [6 marks]
Edexcel M2 Q6
11 marks Standard +0.3
Two railway trucks, \(P\) and \(Q\), of equal mass, are moving towards each other with speeds \(4u\) and \(5u\) respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. [6 marks]
  2. Show that \(e > \frac{1}{9}\). [2 marks]
\(Q\) now hits a fixed buffer and rebounds along the track. \(P\) continues to move with the speed that it had immediately after it collided with \(Q\).
  1. Prove that it is impossible for a further collision between \(P\) and \(Q\) to occur. [3 marks]
Edexcel M2 Q7
11 marks Standard +0.3
A uniform lamina is in the form of a trapezium \(ABCD\), as shown. \(AB\) and \(DC\) are perpendicular to \(BC\). \(AB = 17\) cm, \(BC = 21\) cm and \(CD = 8\) cm. \includegraphics{figure_7}
  1. Find the distances of the centre of mass of the lamina from
    1. \(AB\),
    2. \(BC\). [8 marks]
The lamina is freely suspended from \(C\) and rests in equilibrium.
  1. Find the angle between \(CD\) and the vertical. [3 marks]
Edexcel M2 Q8
15 marks Moderate -0.3
A stone, of mass 1.5 kg, is projected horizontally with speed 4 ms\(^{-1}\) from a height of 7 m above horizontal ground.
  1. Show that the stone travels about 4.78 m horizontally before it hits the ground. [4 marks]
  2. Find the height of the stone above the ground when it has travelled half of this horizontal distance. [4 marks]
  3. Calculate the potential energy lost by the stone as it moves from its point of projection to the ground. [2 marks]
  4. Showing your method clearly, use your answer to part (c) to find the speed with which the stone hits the ground. [3 marks]
  5. State two modelling assumptions that you have made in answering this question. [2 marks]
Edexcel M2 Q1
4 marks Moderate -0.3
A small ball \(A\) is moving with velocity \((7\mathbf{i} + 12\mathbf{j})\) ms\(^{-1}\). It collides in mid-air with another ball \(B\), of mass \(0.4\) kg, moving with velocity \((-\mathbf{i} + 7\mathbf{j})\) ms\(^{-1}\). Immediately after the collision, \(A\) has velocity \((-3\mathbf{i} + 4\mathbf{j})\) ms\(^{-1}\) and \(B\) has velocity \((6.5\mathbf{i} + 13\mathbf{j})\) ms\(^{-1}\). Calculate the mass of \(A\). [4 marks]