Questions — Edexcel M4 (178 questions)

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Edexcel M4 2002 January Q2
8 marks Challenging +1.2
A ball of mass \(m\) is thrown vertically upwards from the ground with an initial speed \(u\). When the speed of the ball is \(v\), the magnitude of the air resistance is \(mkv\), where \(k\) is a positive constant. By modelling the ball as a particle, find, in terms of \(u\), \(k\) and \(g\), the time taken for the ball to reach its greatest height. [8]
Edexcel M4 2002 January Q3
10 marks Challenging +1.2
A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta\), \(\theta < 45°\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.
  1. Show that \(e = \tan^2 \theta\). [6]
  2. Show that the magnitude of the impulse exerted by \(P\) on the plane is \(mu \sec \theta\). [4]
Edexcel M4 2002 January Q4
11 marks Challenging +1.2
A pilot flying an aircraft at a constant speed of 2000 kmh\(^{-1}\) detects an enemy aircraft 100 km away on a bearing of 045°. The enemy aircraft is flying at a constant velocity of 1500 kmh\(^{-1}\) due west. Find
  1. the course, as a bearing to the nearest degree, that the pilot should set up in order to intercept the enemy aircraft,
  2. the time, to the nearest s, that the pilot will take to reach the enemy aircraft. [11]
Edexcel M4 2002 January Q5
12 marks Challenging +1.2
\includegraphics{figure_1} A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal table. The sphere \(S\) collides with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(km\), \(k > 1\), which is at rest on the table. The coefficient of restitution between the spheres is \(e\). Immediately before the spheres collide the direction of motion of \(S\) makes an angle \(\theta\) with the line joining their centres, as shown in Fig. 1. Immediately after the collision the directions of motion of \(S\) and \(T\) are perpendicular.
  1. Show that \(e = \frac{1}{k}\). [6]
Given that \(k = 2\) and that the kinetic energy lost in the collision is one quarter of the initial kinetic energy,
  1. find the value of \(\theta\). [6]
Edexcel M4 2002 January Q6
15 marks Standard +0.8
\includegraphics{figure_2} In a simple model of a shock absorber, a particle \(P\) of mass \(m\) kg is attached to one end of a light elastic horizontal spring. The other end of the spring is fixed at \(A\) and the motion of \(P\) takes place along a fixed horizontal line through \(A\). The spring has natural length \(L\) metres and modulus of elasticity \(2mL\) newtons. The whole system is immersed in a fluid which exerts a resistance on \(P\) of magnitude \(3mv\) newtons, where \(v\) m s\(^{-1}\) is the speed of \(P\) at time \(t\) seconds. The compression of the spring at time \(t\) seconds is \(x\) metres, as shown in Fig. 2.
  1. Show that $$\frac{\text{d}^2 x}{\text{d}t^2} + 3\frac{\text{d}x}{\text{d}t} + 2x = 0.$$ [4]
Given that when \(t = 0\), \(x = 2\) and \(\frac{\text{d}x}{\text{d}t} = -4\),
  1. find \(x\) in terms of \(t\). [8]
  2. Sketch the graph of \(x\) against \(t\). [2]
  3. State, with a reason, whether the model is realistic. [1]
Edexcel M4 2002 January Q7
15 marks Challenging +1.8
\includegraphics{figure_3} A uniform rod \(AB\), of mass \(m\) and length \(2a\), can rotate freely in a vertical plane about a fixed smooth horizontal axis through \(A\). The fixed point \(C\) is vertically above \(A\) and \(AC = 4a\). A light elastic string, of natural length \(2a\) and modulus of elasticity \(\frac{1}{4}mg\), joins \(B\) to \(C\). The rod \(AB\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$-mga[\cos \theta + \sqrt{(5 - 4 \cos \theta)}] + \text{constant}.$$ [9]
  2. Hence determine the values of \(\theta\) for which the system is in equilibrium. [6]
Edexcel M4 2003 January Q1
6 marks Standard +0.8
A boy enters a large horizontal field and sees a friend 100 m due north. The friend is walking in an easterly direction at a constant speed of 0.75 m s\(^{-1}\). The boy can walk at a maximum speed of 1 m s\(^{-1}\). Find the shortest time for the boy to intercept his friend and the bearing on which he must travel to achieve this. [6]
Edexcel M4 2003 January Q2
7 marks Standard +0.8
Boat \(A\) is sailing due east at a constant speed of 10 km h\(^{-1}\). To an observer on \(A\), the wind appears to be blowing from due south. A second boat \(B\) is sailing due north at a constant speed of 14 km h\(^{-1}\). To an observer on \(B\), the wind appears to be blowing from the south west. The velocity of the wind relative to the earth is constant and is the same for both boats. Find the velocity of the wind relative to the earth, stating its magnitude and direction. [7]
Edexcel M4 2003 January Q3
11 marks Challenging +1.2
A small pebble of mass \(m\) is placed in a viscous liquid and sinks vertically from rest through the liquid. When the speed of the pebble is \(v\) the magnitude of the resistance due to the liquid is modelled as \(mkv^2\), where \(k\) is a positive constant. Find the speed of the pebble after it has fallen a distance \(D\) through the liquid. [11]
Edexcel M4 2003 January Q4
16 marks Challenging +1.8
\includegraphics{figure_1} Figure 1 shows a uniform rod \(AB\), of mass \(m\) and length \(4a\), resting on a smooth fixed sphere of radius \(a\). A light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{1}{4}mg\), has one end attached to the lowest point \(C\) of the sphere and the other end attached to \(A\). The points \(A\), \(B\) and \(C\) lie in a vertical plane with \(\angle BAC = 2\theta\), where \(\theta < \frac{\pi}{4}\). Given that \(AC\) is always horizontal,
  1. show that the potential energy of the system is $$\frac{mga}{8}(16\sin 2\theta + 3\cot^2 \theta - 6\cot \theta) + \text{constant}.$$ [7]
  2. show that there is a value of \(\theta\) for which the system is in equilibrium such that \(0.535 < \theta < 0.545\). [6]
  3. Determine whether this position of equilibrium is stable or unstable. [3]
Edexcel M4 2003 January Q5
17 marks Challenging +1.3
A particle \(P\) moves in a straight line. At time \(t\) seconds its displacement from a fixed point \(O\) on the line is \(x\) metres. The motion of \(P\) is modelled by the differential equation $$\frac{\text{d}^2 x}{\text{d}t^2} + 2\frac{\text{d}x}{\text{d}t} + 2x = 12\cos 2t - 6\sin 2t.$$ When \(t = 0\), \(P\) is at rest at \(O\).
  1. Find, in terms of \(t\), the displacement of \(P\) from \(O\). [11]
  2. Show that \(P\) comes to instantaneous rest when \(t = \frac{\pi}{4}\). [2]
  3. Find, in metres to 3 significant figures, the displacement of \(P\) from \(O\) when \(t = \frac{\pi}{4}\). [2]
  4. Find the approximate period of the motion for large values of \(t\). [2]
Edexcel M4 2003 January Q6
18 marks Challenging +1.8
\includegraphics{figure_2} A small ball \(Q\) of mass \(2m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac{13}{12}u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d_1\) from \(B\), and \(BC\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d_2\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(BC\), as shown in Fig. 2, where \(\tan \alpha = \frac{5}{12}\). The line of centres of \(P\) and \(Q\) is parallel to \(BC\). After the collision \(Q\) moves towards \(C\) with speed \(\frac{5}{4}u\).
  1. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(CB\) are \(\frac{1}{4}u\) and \(\frac{5}{12}u\) respectively. [4]
  2. Find the coefficient of restitution between \(P\) and \(Q\). [2]
  3. Show that when \(Q\) reaches \(C\), \(P\) is at a distance \(\frac{4}{5}d_1\) from \(W\). [3]
For each collision between a ball and a wall the coefficient of restitution is \(\frac{1}{2}\). Given that the balls collide with each other again,
  1. show that the time between the two collisions of the balls is \(\frac{15d_1}{u}\). [4]
  2. find the ratio \(d_1 : d_2\). [5]
Edexcel M4 2004 January Q1
5 marks Standard +0.3
A particle \(P\) of mass 3 kg moves in a straight line on a smooth horizontal plane. When the speed of \(P\) is \(v\) m s\(^{-1}\), the resultant force acting on \(P\) is a resistance to motion of magnitude \(2v\) N. Find the distance moved by \(P\) while slowing down from 5 m s\(^{-1}\) to 2 m s\(^{-1}\). [5]
Edexcel M4 2004 January Q2
13 marks Standard +0.8
\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of \(A\) is 2.5 m s\(^{-1}\) and the speed of \(B\) is 1.3 m s\(^{-1}\). When they collide the line joining their centres makes an angle \(\alpha\) with the direction of motion of \(A\) and an angle \(\beta\) with the direction of motion of \(B\), where \(\tan \alpha = \frac{4}{3}\) and \(\tan \beta = \frac{12}{5}\) as shown in Fig. 1.
  1. Find the components of the velocities of \(A\) and \(B\) perpendicular and parallel to the line of centres immediately before the collision. [4]
The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).
  1. Find, to one decimal place, the speed of each sphere after the collision. [9]
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]