Questions M4 (327 questions)

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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]
Edexcel M4 2007 June Q6
17 marks Challenging +1.8
A small ball is attached to one end of a spring. The ball is modelled as a particle of mass 0.1 kg and the spring is modelled as a light elastic spring \(AB\), of natural length 0.5 m and modulus of elasticity 2.45 N. The particle is attached to the end \(B\) of the spring. Initially, at time \(t = 0\), \(A\) is held at rest and the particle hangs at rest in equilibrium below \(A\) at the point \(E\). The end \(A\) then begins to move along the line of the spring in such a way that, at time \(t\) seconds, \(t \leq 1\), the downward displacement of \(A\) from its initial position is \(2 \sin 2t\) metres. At time \(t\) seconds, the extension of the spring is \(x\) metres and the displacement of the particle below \(E\) is \(y\) metres.
  1. Show, by referring to a simple diagram, that \(y + 0.2 = x + 2 \sin 2t\). [3]
  2. Hence show that \(\frac{d^2y}{dt^2} + 49y = 98 \sin 2t\). [5]
Given that \(y = \frac{98}{45} \sin 2t\) is a particular integral of this differential equation,
  1. find \(y\) in terms of \(t\). [5]
  2. Find the time at which the particle first comes to instantaneous rest. [4]
Edexcel M4 2013 June Q1
13 marks Standard +0.8
A particle \(P\) of mass \(0.5\) kg falls vertically from rest. After \(t\) seconds it has speed \(v\) m s\(^{-1}\). A resisting force of magnitude \(1.5v\) newtons acts on \(P\) as it falls.
  1. Show that \(3v = 9.8(1 - e^{-3t})\). [8]
  2. Find the distance that \(P\) falls in the first two seconds of its motion. [5]
Edexcel M4 2013 June Q2
8 marks Standard +0.3
\includegraphics{figure_1} A river is 50 m wide and flows between two straight parallel banks. The river flows with a uniform speed of \(\frac{2}{3}\) m s\(^{-1}\) parallel to the banks. The points \(A\) and \(B\) are on opposite banks of the river and \(AB\) is perpendicular to both banks of the river, as shown in Figure 1. Keith and Ian decide to swim across the river. The speed relative to the water of both swimmers is \(\frac{10}{9}\) m s\(^{-1}\). Keith sets out from \(A\) and crosses the river in the least possible time, reaching the opposite bank at the point \(C\). Find
  1. the time taken by Keith to reach \(C\), [2]
  2. the distance \(BC\). [2]
Ian sets out from \(A\) and swims in a straight line so as to land on the opposite bank at \(B\).
  1. Find the time taken by Ian to reach \(B\). [4]
Edexcel M4 2013 June Q3
10 marks Challenging +1.8
\includegraphics{figure_2} Two smooth uniform spheres \(A\) and \(B\), of equal radius \(r\), have masses \(3m\) and \(2m\) respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision they are moving with speeds \(u\) and \(2u\) respectively. The centres of the spheres are moving towards each other along parallel paths at a distance \(1.6r\) apart, as shown in Figure 2. The coefficient of restitution between the two spheres is \(\frac{1}{6}\). Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision. [10]
Edexcel M4 2013 June Q4
10 marks Challenging +1.2
\includegraphics{figure_3} A small smooth peg \(P\) is fixed at a distance \(d\) from a fixed smooth vertical wire. A particle of mass \(3m\) is attached to one end of a light inextensible string which passes over \(P\). The particle hangs vertically below \(P\). The other end of the string is attached to a small ring \(R\) of mass \(m\), which is threaded on the wire, as shown in Figure 3.
  1. Show that when \(R\) is at a distance \(x\) below the level of \(P\) the potential energy of the system is $$3mg \sqrt{(x^2 + d^2)} - mgx + \text{constant}$$ [4]
  2. Hence find \(x\), in terms of \(d\), when the system is in equilibrium. [3]
  3. Determine the stability of the position of equilibrium. [3]
Edexcel M4 2013 June Q5
8 marks Standard +0.8
A coastguard ship \(C\) is due south of a ship \(S\). Ship \(S\) is moving at a constant speed of 12 km h\(^{-1}\) on a bearing of 140°. Ship \(C\) moves in a straight line with constant speed \(V\) km h\(^{-1}\) in order to intercept \(S\).
  1. Find, giving your answer to 3 significant figures, the minimum possible value for \(V\). [3]
It is now given that \(V = 14\)
  1. Find the bearing of the course that \(C\) takes to intercept \(S\). [5]
Edexcel M4 2013 June Q6
14 marks Challenging +1.3
A particle \(P\) of mass \(m\) kg is attached to the end \(A\) of a light elastic string \(AB\), of natural length \(a\) metres and modulus of elasticity \(9ma\) newtons. Initially the particle and the string lie at rest on a smooth horizontal plane with \(AB = a\) metres. At time \(t = 0\) the end \(B\) of the string is set in motion and moves at a constant speed \(U\) m s\(^{-1}\) in the direction \(AB\). The air resistance acting on \(P\) has magnitude \(6mv\) newtons, where \(v\) m s\(^{-1}\) is the speed of \(P\). At time \(t\) seconds, the extension of the string is \(x\) metres and the displacement of \(P\) from its initial position is \(y\) metres. Show that, while the string is taut,
  1. \(x + y = Ut\) [2]
  2. \(\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 9x = 6U\) [5]
You are given that the general solution of the differential equation in (b) is $$x = (A + Bt)e^{-3t} + \frac{2U}{3}$$ where \(A\) and \(B\) are arbitrary constants.
  1. Find the value of \(A\) and the value of \(B\). [5]
  2. Find the speed of \(P\) at time \(t\) seconds. [2]
Edexcel M4 2013 June Q7
12 marks Challenging +1.8
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane] A small smooth ball of mass \(m\) kg is moving on a smooth horizontal plane and strikes a fixed smooth vertical wall. The plane and the wall intersect in a straight line which is parallel to the vector \(2\mathbf{i} + \mathbf{j}\). The velocity of the ball immediately before the impact is \(b\mathbf{i} + \mathbf{j}\) m s\(^{-1}\), where \(b\) is positive. The velocity of the ball immediately after the impact is \(a(\mathbf{i} + \mathbf{j})\) m s\(^{-1}\), where \(a\) is positive.
  1. Show that the impulse received by the ball when it strikes the wall is parallel to \((-\mathbf{i} + 2\mathbf{j})\). [1]
Find
  1. the coefficient of restitution between the ball and the wall, [8]
  2. the fraction of the kinetic energy of the ball that is lost due to the impact. [3]
Edexcel M4 2014 June Q1
Challenging +1.2
A small smooth ball of mass \(m\) is falling vertically when it strikes a fixed smooth plane which is inclined to the horizontal at an angle \(\alpha\), where \(0° < \alpha < 45°\). Immediately before striking the plane the ball has speed \(u\). Immediately after striking the plane the ball moves in a direction which makes an angle of \(45°\) with the plane. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(m\), \(u\) and \(e\), the magnitude of the impulse of the plane on the ball. (11)
Edexcel M4 2014 June Q2
Standard +0.8
A ship \(A\) is travelling at a constant speed of 30 km h\(^{-1}\) on a bearing of \(050°\). Another ship \(B\) is travelling at a constant speed of \(v\) km h\(^{-1}\) and sets a course to intercept \(A\). At 1400 hours \(B\) is 20 km from \(A\) and the bearing of \(A\) from \(B\) is \(290°\).
  1. Find the least possible value of \(v\). (3)
Given that \(v = 32\),
  1. find the time at which \(B\) intercepts \(A\). (8)
Edexcel M4 2014 June Q3
Challenging +1.2
A small ball of mass \(m\) is projected vertically upwards from a point \(O\) with speed \(U\). The ball is subject to air resistance of magnitude \(mkv\), where \(v\) is the speed of the ball and \(k\) is a positive constant. Find, in terms of \(U\), \(g\) and \(k\), the maximum height above \(O\) reached by the ball. (8)
Edexcel M4 2014 June Q4
Challenging +1.8
A smooth uniform sphere \(S\) is moving on a smooth horizontal plane when it collides obliquely with an identical sphere \(T\) which is at rest on the plane. Immediately before the collision \(S\) is moving with speed \(U\) in a direction which makes an angle of \(60°\) with the line joining the centres of the spheres. The coefficient of restitution between the spheres is \(e\).
  1. Find, in terms of \(e\) and \(U\) where necessary,
    1. the speed and direction of motion of \(S\) immediately after the collision,
    2. the speed and direction of motion of \(T\) immediately after the collision.
    (12)
The angle through which the direction of motion of \(S\) is deflected is \(\delta°\).
  1. Find
    1. the value of \(e\) for which \(\delta\) takes the largest possible value,
    2. the value of \(\delta\) in this case.
    (3)
Edexcel M4 2014 June Q5
Challenging +1.8
\includegraphics{figure_1} A uniform rod \(AB\), of length \(2l\) and mass \(12m\), has its end \(A\) smoothly hinged to a fixed point. One end of a light inextensible string is attached to the other end \(B\) of the rod. The string passes over a small smooth pulley which is fixed at the point \(C\), where \(AC\) is horizontal and \(AC = 2l\). A particle of mass \(m\) is attached to the other end of the string and the particle hangs vertically below \(C\). The angle \(BAC\) is \(\theta\), where \(0 < \theta < \frac{\pi}{2}\), as shown in Figure 1.
  1. Show that the potential energy of the system is $$4mgl\left(\sin\frac{\theta}{2} - 3\sin\theta\right) + \text{constant}$$ (4)
  1. Find the value of \(\theta\) when the system is in equilibrium and determine the stability of this equilibrium position. (10)
Edexcel M4 2014 June Q6
Challenging +1.3
\includegraphics{figure_2} A railway truck of mass \(M\) approaches the end of a straight horizontal track and strikes a buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled as a light horizontal spring \(PQ\), which is fixed at the end \(P\). The spring has a natural length \(a\) and modulus of elasticity \(Mn^2a\), where \(n\) is a positive constant. At time \(t = 0\), the spring has length \(a\) and the truck strikes the end \(Q\) with speed \(U\). A resistive force whose magnitude is \(Mkv\), where \(v\) is the speed of the truck at time \(t\), and \(k\) is a positive constant, also opposes the motion of the truck. At time \(t\), the truck is in contact with the buffer and the compression of the buffer is \(x\).
  1. Show that, while the truck is compressing the buffer $$\frac{\text{d}^2x}{\text{d}t^2} + k\frac{\text{d}x}{\text{d}t} + n^2x = 0$$ (4)
It is given that \(k = \frac{5n}{2}\)
  1. Find \(x\) in terms of \(U\), \(n\) and \(t\). (7)
  1. Find, in terms of \(U\) and \(n\), the greatest value of \(x\). (5)