Questions M4 (327 questions)

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Edexcel M4 2004 January Q3
14 marks Challenging +1.8
\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), each of mass \(2m\) and length \(2L\), are freely jointed at \(A\). The mid-points of the rods are \(D\) and \(E\) respectively. A light inextensible string of length \(s\) is fixed to \(E\) and passes round small, smooth light pulleys at \(D\) and \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs vertically. The points \(A\), \(B\) and \(C\) lie in the same vertical plane with \(B\) and \(C\) on a smooth horizontal surface. The angles \(PAB\) and \(PAC\) are each equal to \(\theta\) (\(\theta > 0\)), as shown in Fig. 2.
  1. Find the length of \(AP\) in terms of \(s\), \(L\) and \(\theta\). [2]
  2. Show that the potential energy \(V\) of the system is given by $$V = 2mgL(3\cos\theta + \sin\theta) + \text{constant}.$$ [4]
  3. Hence find the value of \(\theta\) for which the system is in equilibrium. [4]
  4. Determine whether this position of equilibrium is stable or unstable. [4]
Edexcel M4 2004 January Q4
14 marks Challenging +1.8
A particle \(P\) of mass \(m\) is attached to the mid-point of a light elastic string, of natural length \(2L\) and modulus of elasticity \(2mk^2L\), where \(k\) is a positive constant. The ends of the string are attached to points \(A\) and \(B\) on a smooth horizontal surface, where \(AB = 3L\). The particle is released from rest at the point \(C\), where \(AC = 2L\) and \(ACB\) is a straight line. During the subsequent motion \(P\) experiences air resistance of magnitude \(2mkv\), where \(v\) is the speed of \(P\). At time \(t\), \(AP = 1.5L + x\).
  1. Show that \(\frac{d^2x}{dt^2} + 2k\frac{dx}{dt} + 4k^2x = 0\). [6]
  2. Find an expression, in terms of \(t\), \(k\) and \(L\), for the distance \(AP\) at time \(t\). [8]
Edexcel M4 2004 January Q5
14 marks Challenging +1.2
\includegraphics{figure_3} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point \(X\) with a stationary van of mass 800 kg. After the collision the van came to rest at the point \(A\) having travelled a horizontal distance of 45 m, and the car came to rest at the point \(B\) having travelled a horizontal distance of 21 m. The angle \(AXB\) is 90°. The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.
  1. Find the coefficient of restitution between the car and the van. [5]
The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along \(AX\) and the car to a constant horizontal force of 300 N along \(BX\).
  1. Find the speed of the car immediately before the collision. [9]
Edexcel M4 2004 January Q6
15 marks Standard +0.3
\includegraphics{figure_4} Mary swims in still water at 0.85 m s\(^{-1}\). She swims across a straight river which is 60 m wide and flowing at 0.4 m s\(^{-1}\). She sets off from a point \(A\) on the near bank and lands at a point \(B\), which is directly opposite \(A\) on the far bank, as shown in Fig. 4. Find
  1. the angle between the near bank and the direction in which Mary swims, [3]
  2. the time she takes to cross the river. [3]
\includegraphics{figure_5} A little further downstream a large tree has fallen from the far bank into the river. The river is modelled as flowing at 0.5 m s\(^{-1}\) for a width of 40 m from the near bank, and 0.2 m s\(^{-1}\) for the 20 m beyond this. Nassim swims at 0.85 m s\(^{-1}\) in still water. He swims across the river from a point \(C\) on the near bank. The point \(D\) on the far bank is directly opposite \(C\), as shown in Fig. 5. Nassim swims at the same angle to the near bank as Mary.
  1. Find the maximum distance, downstream from \(CD\), of Nassim during the crossing. [5]
  2. Show that he will land at the point \(D\). [4]
Edexcel M4 2005 January Q1
7 marks Standard +0.8
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors.] Two smooth uniform spheres \(A\) and \(B\) have equal radius but masses \(m\) and \(5m\) respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocities of \(A\) and \(B\) are \((\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and \((-\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\) respectively. Immediately after the collision, the velocity of \(A\) is \((-2\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\).
  1. By considering the impulse on \(A\), find a unit vector parallel to the line joining the centres of the spheres when they collide. [4]
  2. Find the velocity of \(B\) immediately after the collision. [3]
Edexcel M4 2005 January Q2
7 marks Standard +0.3
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively.] A man cycling at a constant speed \(u\) on horizontal ground finds that, when his velocity is \(u\mathbf{j}\) m s\(^{-1}\), the velocity of the wind appears to be \(v(3\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\), where \(v\) is a constant. When the velocity of the man is \(\frac{u}{5}(-3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\), he finds that the velocity of the wind appears to be \(w\mathbf{i}\) m s\(^{-1}\), where \(w\) is a constant.
  1. Show that \(v = \frac{u}{20}\), and find \(w\) in terms of \(u\). [5]
  2. Find, in terms of \(u\), the true velocity of the wind. [2]
Edexcel M4 2005 January Q3
7 marks Standard +0.8
Two ships \(A\) and \(B\) are sailing in the same direction at constant speeds of 12 km h\(^{-1}\) and 16 km h\(^{-1}\) respectively. They are sailing along parallel lines which are 4 km apart. When the distance between the ships is 4 km, \(B\) turns through 30° towards \(A\). Find the shortest distance between the ships in the subsequent motion. [7]
Edexcel M4 2005 January Q4
9 marks Challenging +1.2
A car of mass \(M\) moves along a straight horizontal road. The total resistance to motion of the car is modelled as having constant magnitude \(R\). The engine of the car works at a constant rate \(RU\). Find the time taken for the car to accelerate from a speed of \(\frac{1}{4}U\) to a speed of \(\frac{1}{2}U\). [9]
Edexcel M4 2005 January Q5
10 marks Standard +0.8
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors.] The vector \(\mathbf{n} = (-\frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j})\) and the vector \(\mathbf{p} = (-\frac{4}{5}\mathbf{i} + \frac{3}{5}\mathbf{j})\) are perpendicular unit vectors.
  1. Verify that \(\frac{3}{5}\mathbf{n} + \frac{4}{5}\mathbf{p} = (\mathbf{i} + 3\mathbf{j})\). [2]
A smooth uniform sphere \(S\) of mass 0.5 kg is moving on a smooth horizontal plane when it collides with a fixed vertical wall which is parallel to \(\mathbf{p}\). Immediately after the collision the velocity of \(S\) is \((\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\). The coefficient of restitution between \(S\) and the wall is \(\frac{3}{5}\).
  1. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the velocity of \(S\) immediately before the collision. [5]
  2. Find the energy lost in the collision. [3]
Edexcel M4 2005 January Q6
17 marks Challenging +1.8
\includegraphics{figure_1} A smooth wire \(PMQ\) is in the shape of a semicircle with centre \(O\) and radius \(a\). The wire is fixed in a vertical plane with \(PQ\) horizontal and the mid-point \(M\) of the wire vertically below \(O\). A smooth bead \(B\) of mass \(m\) is threaded on the wire and is attached to one end of a light elastic string. The string has modulus of elasticity \(4mg\) and natural length \(\frac{3}{4}a\). The other end of the string is attached to a fixed point \(P\) which is a distance \(a\) vertically above \(O\), as shown in Fig. 1.
  1. Show that, when \(\angle BFO = \theta\), the potential energy of the system is $$\frac{1}{16}mga(8 \cos \theta - 5)^2 - 2mga \cos^2\theta + \text{constant}.$$ [6]
  2. Hence find the values of \(\theta\) for which the system is in equilibrium. [6]
  3. Determine the nature of the equilibrium at each of these positions. [5]
Edexcel M4 2005 January Q7
18 marks Challenging +1.2
A particle of mass \(m\) is attached to one end \(P\) of a light elastic spring \(PQ\), of natural length \(a\) and modulus of elasticity \(man^2\). At time \(t = 0\), the particle and the spring are at rest on a smooth horizontal table, with the spring straight but unstretched and uncompressed. The end \(Q\) of the spring is then moved in a straight line, in the direction \(PQ\), with constant acceleration \(f\). At time \(t\), the displacement of the particle in the direction \(PQ\) from its initial position is \(x\) and the length of the spring is \((a + y)\).
  1. Show that \(x + y = \frac{1}{2}ft^2\). [2]
  2. Hence show that $$\frac{d^2x}{dt^2} + n^2x = \frac{1}{2}n^2ft^2.$$ [6]
You are given that the general solution of this differential equation is $$x = A\cos nt + B\sin nt + \frac{1}{2}ft^2 - \frac{f}{n^2},$$ where \(A\) and \(B\) are constants.
  1. Find the values of \(A\) and \(B\). [6]
  2. Find the maximum tension in the spring. [4]
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]