Questions — Edexcel M2 (623 questions)

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Edexcel M2 2013 June Q6
12 marks Standard +0.3
\includegraphics{figure_3} A uniform rod \(AB\) has weight 30 N and length 3 m. The rod rests in equilibrium on a rough horizontal peg \(P\) with its end \(A\) on smooth horizontal ground. The rod is in a vertical plane perpendicular to the peg. The rod is inclined at 15° to the ground and the point of contact between the peg and the rod is 45 cm above the ground, as shown in Figure 3.
  1. Show that the normal reaction at \(P\) has magnitude 25 N. [4]
  2. Find the magnitude of the force on the rod at \(A\). [4]
The coefficient of friction between the rod and the peg is \(\mu\).
  1. Find the range of possible values of \(\mu\). [4]
Edexcel M2 2013 June Q7
13 marks Standard +0.3
\includegraphics{figure_4} Two smooth particles \(P\) and \(Q\) have masses \(m\) and \(2m\) respectively. The particles are moving in the same direction in the same straight line, on a smooth horizontal plane, with \(Q\) in front of \(P\). The particles are moving towards a fixed smooth vertical wall which is perpendicular to the direction of motion of the particles, as shown in Figure 4. The speed of \(P\) is \(2u\) and the speed of \(Q\) is \(3u\). The coefficient of restitution between \(Q\) and the wall is \(\frac{1}{3}\). Particle \(Q\) strikes the wall, rebounds and then collides directly with \(P\). The direction of motion of each particle is reversed by this collision. Immediately after this collision the speed of \(P\) is \(v\) and the speed of \(Q\) is \(w\).
  1. Show that \(v = 2w\). [5]
The total kinetic energy of \(P\) and \(Q\) immediately after they collide is half the total kinetic energy of \(P\) and \(Q\) immediately before they collide.
  1. Find the coefficient of restitution between \(P\) and \(Q\). [8]
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]
Edexcel M2 Q2
6 marks Standard +0.3
A stick of mass \(0.75\) kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick and the floor is \(0.6\). Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \includegraphics{figure_2} [6 marks]
Edexcel M2 Q3
7 marks Standard +0.3
An engine of mass \(20\,000\) kg climbs a hill inclined at \(10°\) to the horizontal. The total non-gravitational resistance to its motion has magnitude \(35\,000\) N and the maximum speed of the engine on the hill is \(15\) ms\(^{-1}\).
  1. Find, in kW, the maximum rate at which the engine can work. [4 marks]
  2. Find the maximum speed of the engine when it is travelling on a horizontal track against the same non-gravitational resistance as before. [3 marks]
Edexcel M2 Q4
7 marks Moderate -0.3
Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in the directions due east and due north respectively. A particle \(P\) starts from \(X\), and \(t\) seconds later its position vector relative to \(O\) is \((2t + 4)\mathbf{i} + (kt^2 - 5)\mathbf{j}\).
  1. Find the value of \(k\) if \(P\) takes \(4\) seconds to reach \(Y\). [3 marks]
  2. Show that \(P\) has constant acceleration and find the magnitude and direction of this acceleration. [4 marks]
Edexcel M2 Q5
10 marks Standard +0.8
Three particles \(A\), \(B\) and \(C\), of equal size and each of mass \(m\), are at rest on the same straight line on a smooth horizontal surface. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(e\). \(A\) is projected with speed \(7\) ms\(^{-1}\) and strikes \(B\) directly. \(B\) then collides with \(C\), which starts to move with speed \(4\) ms\(^{-1}\). Calculate the value of \(e\). [10 marks]
Edexcel M2 Q6
11 marks Standard +0.8
A rectangular piece of cardboard \(ABCD\), measuring \(30\) cm by \(12\) cm, has a semicircle of radius \(5\) cm removed from it as shown. \includegraphics{figure_6}
  1. Calculate the distances of the centre of mass of the remaining piece of cardboard from \(AB\) and from \(BC\). [7 marks]
The remaining cardboard is suspended from \(A\) and hangs in equilibrium.
  1. Find the angle made by \(AB\) with the vertical. [4 marks]
Edexcel M2 Q7
15 marks Standard +0.3
A rocket is fired from a fixed point \(O\). During the first phase of its motion its velocity, \(v\) ms\(^{-1}\), is given at time \(t\) seconds after firing by the formula $$v = pt^2 + qt.$$ \(5\) seconds after firing, the rocket is travelling at \(500\) ms\(^{-1}\). \(30\) seconds after firing, the rocket is travelling at \(12\,000\) ms\(^{-1}\).
  1. Find the constants \(p\) and \(q\). [4 marks]
  2. Sketch a velocity-time graph for the rocket for \(0 \leq t \leq 30\). [2 marks]
  3. Find the initial acceleration of the rocket. [2 marks]
  4. Find the distance of the rocket from \(O\) \(30\) seconds after firing. [4 marks]
From time \(t = 30\) onwards, the rocket maintains a constant speed of \(12\,000\) ms\(^{-1}\).
  1. Find the average speed of the rocket during its first \(50\) seconds of motion. [3 marks]