Questions — Edexcel M4 (178 questions)

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Edexcel M4 2006 January Q1
7 marks Standard +0.3
A particle \(P\) of mass \(0.5\) kg is released from rest at time \(t = 0\) and falls vertically through a liquid. The motion of \(P\) is resisted by a force of magnitude \(2v\) N, where \(v\) m s\(^{-1}\) is the speed of \(v\) at time \(t\) seconds.
  1. Show that \(5 \frac{\mathrm{d}v}{\mathrm{d}t} = 49 - 20v\). [2]
  2. Find the speed of \(P\) when \(t = 1\). [5]
Edexcel M4 2006 January Q2
11 marks Challenging +1.2
A small smooth sphere \(S\) of mass \(m\) is attached to one end of a light inextensible string of length \(2a\). The other end of the string is attached to a fixed point \(A\) which is at a distance \(a\sqrt{3}\) from a smooth vertical wall. The sphere \(S\) hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed \(\sqrt{\left(\frac{37ga}{5}\right)}\).
  1. Show that \(S\) strikes the wall with speed \(\sqrt{\left(\frac{27ga}{5}\right)}\). [4] Given that the loss in kinetic energy due to the impact with the wall is \(\frac{3mga}{5}\),
  2. find the coefficient of restitution between \(S\) and the wall. [7]
Edexcel M4 2006 January Q3
12 marks Standard +0.8
Two ships \(P\) and \(Q\) are moving with constant velocity. At 3 p.m., \(P\) is 20 km due north of \(Q\) and is moving at 16 km h\(^{-1}\) due west. To an observer on ship \(P\), ship \(Q\) appears to be moving on a bearing of \(030°\) at 10 km h\(^{-1}\). Find
    1. the speed of \(Q\),
    2. the direction in which \(Q\) is moving, giving your answer as a bearing to the nearest degree,
    [6]
  2. the shortest distance between the ships, [3]
  3. the time at which the two ships are closest together. [3]
Edexcel M4 2006 January Q4
12 marks Standard +0.8
A particle \(P\) of mass \(m\) is suspended from a fixed point by a light elastic spring. The spring has natural length \(a\) and modulus of elasticity \(2m\omega^2a\), where \(\omega\) is a positive constant. At time \(t = 0\) the particle is projected vertically downwards with speed \(U\) from its equilibrium position. The motion of the particle is resisted by a force of magnitude \(2m\omega v\), where \(v\) is the speed of the particle. At time \(t\), the displacement of \(P\) downwards from its equilibrium position is \(x\).
  1. Show that \(\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\omega \frac{\mathrm{d}x}{\mathrm{d}t} + 2\omega^2x = 0\). [5] Given that the solution of this differential equation is \(x = e^{-\omega t}(A \cos \omega t + B \sin \omega t)\), where \(A\) and \(B\) are constants,
  2. find \(A\) and \(B\). [4]
  3. Find an expression for the time at which \(P\) first comes to rest. [3]
Edexcel M4 2006 January Q5
16 marks Challenging +1.2
Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(km\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of \(60°\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Show that the speed of \(B\) immediately after the collision is \(\frac{3u}{4(k + 1)}\). [6] Immediately after the collision the direction of motion of \(A\) makes an angle arctan \((2\sqrt{3})\) with the direction of motion of \(B\).
  2. Show that \(k = \frac{1}{2}\). [6]
  3. Find the loss of kinetic energy due to the collision. [4]
Edexcel M4 2006 January Q6
17 marks Challenging +1.8
\includegraphics{figure_1} A smooth wire with ends \(A\) and \(B\) is in the shape of a semi-circle of radius \(a\). The mid-point of \(AB\) is \(O\) and is fixed in a vertical plane and hangs below \(AB\) which is horizontal. A small ring \(R\), of mass \(m\sqrt{2}\), is threaded on the wire and is attached to two light inextensible strings. The other end of each string is attached to a particle of mass \(\frac{3m}{2}\). The particles hang vertically under gravity, as shown in Figure 1.
  1. Show that, when the radius \(OR\) makes an angle \(2\theta\) with the vertical, the potential energy, \(V\), of the system is given by $$V = \sqrt{2}mga(3 \cos \theta - \cos 2\theta) + \text{constant}.$$ [7]
  2. Find the values of \(\theta\) for which the system is in equilibrium. [6]
  3. Determine the stability of the position of equilibrium for which \(\theta > 0\). [4]
Edexcel M4 2005 June Q1
7 marks Standard +0.3
A small smooth ball of mass \(\frac{1}{2}\) kg is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{1}{3}\). Immediately before striking the plane the ball has speed 10 m s\(^{-1}\). The coefficient of restitution between ball and plane is \(\frac{1}{2}\). Find
  1. the speed, to 3 significant figures, of the ball immediately after the impact, [5]
  2. the magnitude of the impulse received by the ball as it strikes the plane. [2]
Edexcel M4 2005 June Q2
5 marks Standard +0.3
A cyclist \(P\) is cycling due north at a constant speed of 20 km h\(^{-1}\). At 12 noon another cyclist \(Q\) is due west of \(P\). The speed of \(Q\) is constant at 10 km h\(^{-1}\). Find the course which \(Q\) should set in order to pass as close to \(P\) as possible, giving your answer as a bearing. [5]
Edexcel M4 2005 June Q3
11 marks Challenging +1.2
\includegraphics{figure_1} A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\). Show that \((1-e) \tan \beta = 2 \tan \alpha\). [11]
Edexcel M4 2005 June Q4
11 marks Standard +0.8
A lorry of mass \(M\) is moving along a straight horizontal road. The engine produces a constant driving force of magnitude \(F\). The total resistance to motion is modelled as having magnitude \(kv^2\), where \(k\) is a constant, and \(v\) is the speed of the lorry. Given the lorry moves with constant speed \(V\),
  1. show that \(V = \sqrt{\frac{F}{k}}\). [2]
Given instead that the lorry starts from rest,
  1. show that the distance travelled by the lorry in attaining a speed of \(\frac{1}{2}V\) is $$\frac{M}{2k}\ln\left(\frac{4}{3}\right).$$ [9]
Edexcel M4 2005 June Q5
12 marks Challenging +1.8
A non-uniform rod \(BC\) has mass \(m\) and length \(3l\). The centre of mass of the rod is at distance \(l\) from \(B\). The rod can turn freely about a fixed smooth horizontal axis through \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\frac{mg}{6}\), is attached to \(C\). The other end of the string is attached to a point \(P\) which is at a height \(3l\) vertically above \(B\).
  1. Show that, while the string is stretched, the potential energy of the system is $$mgl(\cos^2 \theta - \cos \theta) + \text{constant},$$ where \(\theta\) is the angle between the string and the downward vertical and \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\). [6]
  2. Find the values of \(\theta\) for which the system is in equilibrium with the string stretched. [6]
Edexcel M4 2005 June Q6
12 marks Challenging +1.2
A ship \(A\) has maximum speed 30 km h\(^{-1}\). At time \(t = 0\), \(A\) is 70 km due west of \(B\) which is moving at a constant speed of 36 km h\(^{-1}\) on a bearing of 300°. Ship \(A\) moves on a straight course at a constant speed and intercepts \(B\). The course of \(A\) makes an angle \(\theta\) with due north.
  1. Show that \(-\arctan \frac{4}{3} \leq \theta \leq \arctan \frac{4}{3}\). [7]
  2. Find the least time for \(A\) to intercept \(B\). [5]
Edexcel M4 2005 June Q7
17 marks Challenging +1.8
A light elastic string, of natural length \(a\) and modulus of elasticity \(5ma\omega^2\), lies unstretched along a straight line on a smooth horizontal plane. A particle of mass \(m\) is attached to one end of the spring. At time \(t = 0\), the other end of the spring starts to move with constant speed \(U\) along the line of the spring and away from the particle. As the particle moves along the plane it is subject to a resistance of magnitude \(2m\omega v\), where \(v\) is its speed. At time \(t\), the extension of the spring is \(x\) and the displacement of the particle from its initial position is \(y\). Show that
  1. \(x + y = Ut\), [2]
  2. \(\frac{d^2x}{dt^2} + 2\omega \frac{dx}{dt} + 5\omega^2 x = 2\omega U\). [7]
  1. Find \(x\) in terms of \(\omega\), \(U\) and \(t\). [8]
Edexcel M4 2006 June Q1
5 marks Standard +0.3
At noon, a boat \(P\) is on a bearing of \(120°\) from boat \(Q\). Boat \(P\) is moving due east at a constant speed of \(12\) km h\(^{-1}\). Boat \(Q\) is moving in a straight line with a constant speed of \(15\) km h\(^{-1}\) on a course to intercept \(P\). Find the direction of motion of \(Q\), giving your answer as a bearing. [5]
Edexcel M4 2006 June Q2
6 marks Standard +0.3
A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a fixed smooth vertical wall. Immediately before the collision, the speed of \(S\) is \(U\) and its direction of motion makes an angle \(\alpha\) with the wall. The coefficient of restitution between \(S\) and the wall is \(e\). Find the kinetic energy of \(S\) immediately after the collision. [6]
Edexcel M4 2006 June Q3
10 marks Standard +0.3
A cyclist \(C\) is moving with a constant speed of \(10\) m s\(^{-1}\) due south. Cyclist \(D\) is moving with a constant speed of \(16\) m s\(^{-1}\) on a bearing of \(240°\).
  1. Show that the magnitude of the velocity of \(C\) relative to \(D\) is \(14\) m s\(^{-1}\). [3]
At \(2\) pm, \(D\) is \(4\) km due east of \(C\).
  1. Find
    1. the shortest distance between \(C\) and \(D\) during the subsequent motion,
    2. the time, to the nearest minute, at which this shortest distance occurs.
    [7]
Edexcel M4 2006 June Q4
12 marks Challenging +1.2
\includegraphics{figure_1} A uniform rod \(PQ\) has mass \(m\) and length \(2l\). A small smooth light ring is fixed to the end \(P\) of the rod. This ring is threaded on to a fixed horizontal smooth straight wire. A second small smooth light ring \(R\) is threaded on to the wire and is attached by a light elastic string, of natural length \(l\) and modulus of elasticity \(kmg\), to the end \(Q\) of the rod, where \(k\) is a constant.
  1. Show that, when the rod \(PQ\) makes an angle \(\theta\) with the vertical, where \(0 < \theta \leq \frac{\pi}{3}\), and \(Q\) is vertically below \(R\), as shown in Figure 1, the potential energy of the system is $$mgl[2k\cos^2\theta - (2k + 1)\cos\theta] + \text{constant}.$$ [7]
Given that there is a position of equilibrium with \(\theta > 0\),
  1. show that \(k > \frac{1}{2}\). [5]
Edexcel M4 2006 June Q5
11 marks Standard +0.8
A train of mass \(m\) is moving along a straight horizontal railway line. A time \(t\), the train is moving with speed \(v\) and the resistance to motion has magnitude \(kv\), where \(k\) is a constant. The engine of the train is working at a constant rate \(P\).
  1. Show that, when \(v > 0\), \(mv\frac{dv}{dt} + kv^2 = P\). [3]
When \(t = 0\), the speed of the train is \(\frac{1}{3}\sqrt{\frac{P}{k}}\).
  1. Find, in terms of \(m\) and \(k\), the time taken for the train to double its initial speed. [8]
Edexcel M4 2006 June Q6
14 marks Challenging +1.2
\includegraphics{figure_2} Two small smooth spheres \(A\) and \(B\), of equal size and of mass \(m\) and \(2m\) respectively, are moving initially with the same speed \(U\) on a smooth horizontal floor. The spheres collide when their centres are on a line \(L\). Before the collision the spheres are moving towards each other, with their directions of motion perpendicular to each other and each inclined at an angle of \(45°\) to the line \(L\), as shown in Figure 2. The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Find the magnitude of the impulse which acts on \(A\) in the collision. [9]
\includegraphics{figure_3} The line \(L\) is parallel to and a distance \(d\) from a smooth vertical wall, as shown in Figure 3.
  1. Find, in terms of \(d\), the distance between the points at which the spheres first strike the wall. [5]
Edexcel M4 2006 June Q7
17 marks Challenging +1.8
\includegraphics{figure_4} A light elastic spring has natural length \(l\) and modulus of elasticity \(4mg\). One end of the spring is attached to a point \(A\) on a plane that is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac{1}{4}\). The particle \(P\) is held at a point \(B\) on the plane where \(B\) is below \(A\) and \(AB = l\), with the spring lying along a line of greatest slope of the plane, as shown in Figure 4. At time \(t = 0\), the particle is projected up the plane towards \(A\) with speed \(\frac{1}{2}\sqrt{gl}\). At time \(t\), the compression of the spring is \(x\).
  1. Show that $$\frac{d^2x}{dt^2} + 4\omega^2x = -g, \text{ where } \omega = \sqrt{\frac{g}{l}}.$$ [6]
  1. Find \(x\) in terms of \(l\), \(\omega\) and \(t\). [7]
  1. Find the distance that \(P\) travels up the plane before first coming to rest. [4]
Edexcel M4 2007 June Q1
10 marks Challenging +1.2
A small ball is moving on a horizontal plane when it strikes a smooth vertical wall. The coefficient of restitution between the ball and the wall is \(e\). Immediately before the impact the direction of motion of the ball makes an angle of \(60°\) with the wall. Immediately after the impact the direction of motion of the ball makes an angle of \(30°\) with the wall.
  1. Find the fraction of the kinetic energy of the ball which is lost in the impact. [6]
  2. Find the value of \(e\). [4]
Edexcel M4 2007 June Q2
10 marks Standard +0.3
A lorry of mass \(M\) moves along a straight horizontal road against a constant resistance of magnitude \(R\). The engine of the lorry works at a constant rate \(RU\), where \(U\) is a constant. At time \(t\), the lorry is moving with speed \(v\).
  1. Show that \(Mv\frac{dv}{dt} = R(U - v)\). [3]
At time \(t = 0\), the lorry has speed \(\frac{1}{4}U\) and the time taken by the lorry to attain a speed of \(\frac{3}{4}U\) is \(\frac{kMU}{R}\), where \(k\) is a constant.
  1. Find the exact value of \(k\). [7]
Edexcel M4 2007 June Q3
12 marks Challenging +1.2
\includegraphics{figure_1} A framework consists of two uniform rods \(AB\) and \(BC\), each of mass \(m\) and length \(2a\), joined at \(B\). The mid-points of the rods are joined by a light rod of length \(a\sqrt{2}\), so that angle \(ABC\) is a right angle. The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis passes through the point \(A\) and is perpendicular to the plane of the framework. The angle between the rod \(AB\) and the downward vertical is denoted by \(\theta\), as shown in Fig. 1.
  1. Show that the potential energy of the framework is $$-mga(3 \cos \theta + \sin \theta) + \text{constant}.$$ [4]
  2. Find the value of \(\theta\) when the framework is in equilibrium, with \(B\) below the level of \(A\). [4]
  3. Determine the stability of this position of equilibrium. [4]
Edexcel M4 2007 June Q4
13 marks Challenging +1.2
At 12 noon, ship \(A\) is 20 km from ship \(B\), on a bearing of \(300°\). Ship \(A\) is moving at a constant speed of 15 km h\(^{-1}\) on a bearing of \(070°\). Ship \(B\) moves in a straight line with constant speed \(V\) km h\(^{-1}\) and intercepts \(A\).
  1. Find, giving your answer to 3 significant figures, the minimum possible for \(V\). [3]
It is now given that \(V = 13\).
  1. Explain why there are two possible times at which ship \(B\) can intercept ship \(A\). [2]
  2. Find, giving your answer to the nearest minute, the earlier time at which ship \(B\) can intercept ship \(A\). [8]
Edexcel M4 2007 June Q5
13 marks Challenging +1.2
A smooth uniform sphere \(A\) has mass \(2m\) kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(m\) kg. The spheres are moving on a smooth horizontal plane when they collide. At the instant of collision the line joining the centres of the spheres is parallel to \(\mathbf{j}\). Immediately after the collision, the velocity of \(A\) is \((3\mathbf{i} - \mathbf{j})\) m s\(^{-1}\) and the velocity of \(B\) is \((2\mathbf{i} + \mathbf{j})\) m s\(^{-1}\). The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Find the velocities of the two spheres immediately before the collision. [7]
  2. Find the magnitude of the impulse in the collision. [2]
  3. Find, to the nearest degree, the angle through which the direction of motion of \(A\) is deflected by the collision. [4]