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Edexcel M4 2006 January Q3
12 marks Challenging +1.2
3. 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 \(16 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) due west. To an observer on ship \(P\), ship \(Q\) appears to be moving on a bearing of \(030 ^ { \circ }\) at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find
    1. the speed of \(Q\),
    2. the direction in which \(Q\) is moving, giving your answer as a bearing to the nearest degree,
  2. the shortest distance between the ships,
  3. the time at which the two ships are closest together.
    (3)
Edexcel M4 2006 January Q4
12 marks Standard +0.3
4. 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 \(2 m \omega ^ { 2 } a\), 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 \(2 m \omega v\), 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 \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 \omega ^ { 2 } x = 0\). Given that the solution of this differential equation is \(x = \mathrm { e } ^ { - \omega t } ( A \cos \omega t + B \sin \omega t )\), where \(A\) and \(B\) are constants,
  2. find \(A\) and \(B\).
  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
5. Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(k m\). 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 \(60 ^ { \circ }\) 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 \(\frac { 3 u } { 4 ( k + 1 ) }\).
    (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 \(k = \frac { 1 } { 2 }\).
  3. Find the loss of kinetic energy due to the collision.
    (4) \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fe647e21-9c4f-4035-b28f-b12a00087692-4_515_1077_301_502}
    \end{figure} A smooth wire with ends \(A\) and \(B\) is in the shape of a semi-circle of radius \(a\). The mid-point of \(A B\) is \(O\). The wire is fixed in a vertical plane and hangs below \(A B\) 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 \(\frac { 3 m } { 2 }\). The particles hang vertically under gravity, as shown in Figure 1.
  4. Show that, when the radius \(O R\) makes an angle \(2 \theta\) with the vertical, the potential energy, \(V\), of the system is given by $$V = \sqrt { } 2 m g a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
  5. Find the values of \(\theta\) for which the system is in equilibrium.
  6. Determine the stability of the position of equilibrium for which \(\theta > 0\).
Edexcel M4 2002 June Q1
9 marks Standard +0.8
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-2_450_1417_391_339}
\end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 2 kg and sphere \(B\) has mass 3 kg . The spheres collide and at the instant of collision the line joining their centres is parallel to \(\mathbf { i }\). Before the collision \(A\) has velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision it has velocity \(( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Before the collision the velocity of \(B\) makes an angle \(\alpha\) with the line of centres, as shown in Fig. 1, where \(\tan \alpha = 2\). The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) before the collision.
(9)
Edexcel M4 2002 June Q2
10 marks Challenging +1.2
2. Ship \(A\) is steaming on a bearing of \(060 ^ { \circ }\) at \(30 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and at 9 a.m. it is 20 km due west of a second ship \(B\). Ship \(B\) steams in a straight line.
  1. Find the least speed of \(B\) if it is to intercept \(A\). Given that the speed of \(B\) is \(24 \mathrm {~km} \mathrm {~h} ^ { - 1 }\),
  2. find the earliest time at which it can intercept \(A\).
Edexcel M4 2002 June Q3
12 marks Standard +0.8
3. The engine of a car of mass 800 kg works at a constant rate of 32 kW . The car travels along a straight horizontal road and the resistance to motion of the car is proportional to the speed of the car. The car starts from rest and \(t\) seconds later it has a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that $$800 v \frac { \mathrm {~d} v } { \mathrm {~d} t } = 32000 - k v ^ { 2 } , \text { where } k \text { is a positive constant. }$$ Given that the limiting speed of the car is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find
  2. the value of \(k\),
  3. \(v\) in terms of \(t\).
Edexcel M4 2002 June Q4
13 marks Challenging +1.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-3_424_422_1181_844}
\end{figure} Four identical uniform rods, each of mass \(m\) and length \(2 a\), are freely jointed to form a rhombus \(A B C D\). The rhombus is suspended from \(A\) and is prevented from collapsing by an elastic string which joins \(A\) to \(C\), with \(\angle B A D = 2 \theta , 0 \leq \theta \leq \frac { 1 } { 3 } \pi\), as shown in Fig. 2. The natural length of the elastic string is \(2 a\) and its modulus of elasticity is \(4 m g\).
  1. Show that the potential energy, \(V\), of the system is given by $$V = 4 m g a \left[ ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \cos \theta \right] + \text { constant } .$$
  2. Hence find the non-zero value of \(\theta\) for which the system is in equilibrium.
  3. Determine whether this position of equilibrium is stable or unstable.
Edexcel M4 2002 June Q5
14 marks Challenging +1.2
5. At time \(t = 0\) particles \(P\) and \(Q\) start simultaneously from points which have position vectors \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and \(( - \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\) respectively, relative to a fixed origin \(O\). The velocities of \(P\) and \(Q\) are \(( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively.
  1. Show that \(P\) and \(Q\) collide and find the position vector of the point at which they collide. A third particle \(R\) moves in such a way that its velocity relative to \(P\) is parallel to the vector ( \(- 5 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) ) and its velocity relative to \(Q\) is parallel to the vector \(( - 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\). Given that all three particles collide simultaneously, find
    1. the velocity of \(R\),
    2. the position vector of \(R\) at time \(t = 0\).
Edexcel M4 2002 June Q6
17 marks Challenging +1.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-4_244_1264_1314_382}
\end{figure} A particle \(P\) of mass 2 kg is attached to the mid-point of a light elastic spring of natural length 2 m and modulus of elasticity 4 N . One end \(A\) of the elastic spring is attached to a fixed point on a smooth horizontal table. The spring is then stretched until its length is 4 m and its other end \(B\) is held at a point on the table where \(A B = 4 \mathrm {~m}\). At time \(t = 0 , P\) is at rest on the table at the point \(O\) where \(A O = 2 \mathrm {~m}\), as shown in Fig. 3. The end \(B\) is now moved on the table in such a way that \(A O B\) remains a straight line. At time \(t\) seconds, \(A B = \left( 4 + \frac { 1 } { 2 } \sin 4 t \right) \mathrm { m }\) and \(A P = ( 2 + x ) \mathrm { m }\).
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 x = \sin 4 t$$
  2. Hence find the time when \(P\) first comes to instantaneous rest. END
Edexcel M4 2003 June Q1
8 marks Standard +0.8
  1. A wooden ball of mass 0.01 kg falls vertically into a pond of water. The speed of the ball as it enters the water is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the ball is \(x\) metres below the surface of the water and moving downwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the water provides a resistance of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) and an upward buoyancy force of magnitude 0.158 N .
    1. Show that, while the ball is moving downwards,
    $$- 2 v ^ { 2 } - 6 = v \frac { \mathrm {~d} v } { \mathrm {~d} x }$$
  2. Hence find, to 3 significant figures, the greatest distance below the surface of the water reached by the ball.
Edexcel M4 2003 June Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{47e1d96b-4582-4324-a946-66989a2c66fc-2_455_1084_1112_487}
\end{figure} A man, who rows at a speed \(v\) through still water, rows across a river which flows at a speed \(u\). The man sets off from the point \(A\) on one bank and wishes to land at the point \(B\) on the opposite bank, where \(A B\) is perpendicular to both banks, as shown in Fig. 1.
  1. Show that, for this to be possible, \(v > u\). Given that \(v < u\) and that he rows from \(A\) so as to reach a point \(C\), on the opposite bank, which is as close to \(B\) as possible,
  2. find, in terms of \(u\) and \(v\), the ratio of \(B C\) to the width of the river.
    (5)
Edexcel M4 2003 June Q3
9 marks Challenging +1.2
3. A man walks due north at a constant speed \(u\) and the wind seems to him to be blowing from the direction \(30 ^ { \circ }\) east of north. On his return journey, when he is walking at the same speed \(u\) due south, the wind seems to him to be blowing from the direction \(30 ^ { \circ }\) south of east. Assuming that the velocity, \(\mathbf { w }\), of the wind relative to the earth is constant, find
  1. the magnitude of \(\mathbf { w }\), in terms of \(u\),
  2. the direction of \(\mathbf { w }\).
Edexcel M4 2003 June Q4
15 marks Challenging +1.2
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{47e1d96b-4582-4324-a946-66989a2c66fc-3_581_826_801_648}
\end{figure} A uniform rod \(A B\), of length \(2 a\) and mass \(8 m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 5 } \mathrm { mg }\), is fixed to \(B\). The other end of the string is attached to a small ring which is free to slide on a smooth straight horizontal wire which is fixed in the same vertical plane as \(A B\) at a height 7a vertically above \(A\). The rod \(A B\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 2.
  1. Show that the potential energy \(V\) of the system is given by $$V = \frac { 8 } { 5 } m g a \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant. }$$
  2. Hence find the values of \(\theta , 0 \leq \theta \leq \pi\), for which the system is in equilibrium.
  3. Determine the nature of these positions of equilibrium.
Edexcel M4 2003 June Q5
17 marks Challenging +1.2
5. A light elastic string, of natural length \(2 a\) and modulus of elasticity \(m g\), has a particle \(P\) of mass \(m\) attached to its mid-point. One end of the string is attached to a fixed point \(A\) and the other end is attached to a fixed point \(B\) which is at a distance \(4 a\) vertically below \(A\).
  1. Show that \(P\) hangs in equilibrium at the point \(E\) where \(A E = \frac { 5 } { 2 } a\). The particle \(P\) is held at a distance \(3 a\) vertically below \(A\) and is released from rest at time \(t = 0\). When the speed of the particle is \(v\), there is a resistance to motion of magnitude \(2 m k v\), where \(k = \sqrt { } \left( \frac { g } { a } \right)\). At time \(t\) the particle is at a distance \(\left( \frac { 5 } { 2 } a + x \right)\) from \(A\).
  2. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 k ^ { 2 } x = 0$$
  3. Hence find \(x\) in terms of \(t\).
Edexcel M4 2003 June Q6
18 marks Challenging +1.2
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{47e1d96b-4582-4324-a946-66989a2c66fc-5_652_725_296_620}
\end{figure} A small smooth uniform sphere \(S\) is at rest on a smooth horizontal floor at a distance \(d\) from a straight vertical wall. An identical sphere \(T\) is projected along the floor with speed \(U\) towards \(S\) and in a direction which is perpendicular to the wall. At the instant when \(T\) strikes \(S\) the line joining their centres makes an angle \(\alpha\) with the wall, as shown in Fig. 3. Each sphere is modelled as having negligible diameter in comparison with \(d\). The coefficient of restitution between the spheres is \(e\).
  1. Show that the components of the velocity of \(T\) after the impact, parallel and perpendicular to the line of centres, are \(\frac { 1 } { 2 } U ( 1 - e ) \sin \alpha\) and \(U \cos \alpha\) respectively.
  2. Show that the components of the velocity of \(T\) after the impact, parallel and perpendicular to the wall, are \(\frac { 1 } { 2 } U ( 1 + e ) \cos \alpha \sin \alpha\) and \(\frac { 1 } { 2 } U \left[ 2 - ( 1 + e ) \sin ^ { 2 } \alpha \right]\) respectively. The spheres \(S\) and \(T\) strike the wall at the points \(A\) and \(B\) respectively.
    Given that \(e = \frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\),
  3. find, in terms of \(d\), the distance \(A B\). \section*{END}
Edexcel M4 2004 June Q1
6 marks Standard +0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.]
An aeroplane makes a journey from a point \(P\) to a point \(Q\) which is due east of \(P\). The wind velocity is \(w ( \cos \theta \mathbf { i } + \sin \theta \mathbf { j } )\), where \(w\) is a positive constant. The velocity of the aeroplane relative to the wind is \(v ( \cos \phi \mathbf { i } - \sin \phi \mathbf { j } )\), where \(v\) is a constant and \(v > w\). Given that \(\theta\) and \(\phi\) are both acute angles,
  1. show that \(v \sin \phi = w \sin \theta\),
  2. find, in terms of \(v , w\) and \(\theta\), the speed of the aeroplane relative to the ground.
Edexcel M4 2004 June Q2
11 marks Challenging +1.2
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4eb9c38d-66f4-40ba-b7cf-2c2bd19ad087-2_491_826_947_623}
\end{figure} A smooth uniform sphere \(P\) is at rest on a smooth horizontal plane, when it is struck by an identical sphere \(Q\) moving on the plane. Immediately before the impact, the line of motion of the centre of \(Q\) is tangential to the sphere \(P\), as shown in Fig. 1. The direction of motion of \(Q\) is turned through \(30 ^ { \circ }\) by the impact. Find the coefficient of restitution between the spheres.
Edexcel M4 2004 June Q3
11 marks Standard +0.8
3. At noon, two boats \(A\) and \(B\) are 6 km apart with \(A\) due east of \(B\). Boat \(B\) is moving due north at a constant speed of \(13 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Boat \(A\) is moving with constant speed \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and sets a course so as to pass as close as possible to boat \(B\). Find
  1. the direction of motion of \(A\), giving your answer as a bearing,
  2. the time when the boats are closest,
  3. the shortest distance between the boats.
Edexcel M4 2004 June Q4
15 marks Challenging +1.2
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4eb9c38d-66f4-40ba-b7cf-2c2bd19ad087-3_506_967_339_608}
\end{figure} A uniform rod \(P Q\), of length \(2 a\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through the end \(P\). The end \(Q\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { m g } { 2 \sqrt { 3 } }\). The other end of the string is attached to a fixed point \(O\), where \(O P\) is horizontal and \(O P = 2 a\), as shown in Fig. 2. \(\angle O P Q\) is denoted by \(2 \theta\).
  1. Show that, when the string is taut, the potential energy of the system is $$- \frac { m g a } { \sqrt { 3 } } ( 2 \cos 2 \theta + \sqrt { 3 } \sin 2 \theta + 2 \sin \theta ) + \text { constant } .$$
  2. Verify that there is a position of equilibrium at \(\theta = \frac { \pi } { 6 }\).
  3. Determine whether this is a position of stable equilibrium.
Edexcel M4 2004 June Q5
16 marks Standard +0.8
5. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m a k ^ { 2 }\), where \(k\) is a positive constant. The other end of the string is attached to a fixed point \(A\). At time \(t = 0 , P\) is released from rest from a point which is a distance \(2 a\) vertically below \(A\). When \(P\) is moving with speed \(v\), the air resistance has magnitude \(2 m k v\). At time \(t\), the extension of the string is \(x\).
  1. Show that, while the string is taut, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 k ^ { 2 } x = g$$ You are given that the general solution of this differential equation is $$x = \mathrm { e } ^ { - k t } ( C \sin k t + D \cos k t ) + \frac { g } { 2 k ^ { 2 } } , \quad \text { where } C \text { and } D \text { are constants. }$$
  2. Find the value of \(C\) and the value of \(D\). Assuming that the string remains taut,
  3. find the value of \(t\) when \(P\) first comes to rest,
  4. show that \(2 k ^ { 2 } a < g \left( 1 + \mathrm { e } ^ { \pi } \right)\).
Edexcel M4 2004 June Q6
16 marks Challenging +1.8
6. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string and hangs at rest at time \(t = 0\). The other end of the string is then raised vertically by an engine which is working at a constant rate \(k m g\), where \(k > 0\). At time \(t\), the distance of \(P\) above its initial position is \(x\), and \(P\) is moving upwards with speed \(v\).
  1. Show that \(v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = ( k - v ) g\).
  2. Show that \(g x = k ^ { 2 } \ln \left( \frac { k } { k - v } \right) - k v - \frac { 1 } { 2 } v ^ { 2 }\).
  3. Hence, or otherwise, find \(t\) in terms of \(k , v\) and \(g\).
Edexcel M4 2005 June Q1
7 marks Standard +0.3
  1. A small smooth ball of mass \(\frac { 1 } { 2 } \mathrm {~kg}\) is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). Immediately before striking the plane the ball has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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,
    2. the magnitude of the impulse received by the ball as it strikes the plane.
    3. A cyclist \(P\) is cycling due north at a constant speed of \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At 12 noon another cyclist \(Q\) is due west of \(P\). The speed of \(Q\) is constant at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find the course which \(Q\) should set in order to pass as close to \(P\) as possible, giving your answer as a bearing.
      (5)
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6895ccda-84b8-45a5-9524-e5bfc37a2fee-2_437_1232_1174_443}
    \end{figure} 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 \(( 1 - e ) \tan \beta = 2 \tan \alpha\).
Edexcel M4 2005 June Q4
11 marks Standard +0.3
4. 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 \(k v ^ { 2 }\), where \(k\) is a constant, and \(v\) is the speed of the lorry. Given the lorry moves with constant speed \(V\),
  1. show that \(V = \sqrt { \frac { F } { k } }\). Given instead that the lorry starts from rest,
  2. show that the distance travelled by the lorry in attaining a speed of \(\frac { 1 } { 2 } V\) is $$\frac { M } { 2 k } \ln \left( \frac { 4 } { 3 } \right)$$
Edexcel M4 2005 June Q5
12 marks Challenging +1.3
  1. A non-uniform rod \(B C\) has mass \(m\) and length \(3 l\). 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 \(\frac { m g } { 6 }\), is attached to \(C\). The other end of the string is attached to a point \(P\) which is at a height \(3 l\) vertically above \(B\).
    1. Show that, while the string is stretched, the potential energy of the system is
    $$m g l \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant, }$$ where \(\theta\) is the angle between the string and the downward vertical and \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\).
  2. Find the values of \(\theta\) for which the system is in equilibrium with the string stretched.
Edexcel M4 2005 June Q6
12 marks Challenging +1.2
6. A ship \(A\) has maximum speed \(30 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t = 0 , A\) is 70 km due west of \(B\) which is moving at a constant speed of \(36 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(300 ^ { \circ }\). 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 \(- \arctan \frac { 4 } { 3 } \leq \theta \leq \arctan \frac { 4 } { 3 }\).
  2. Find the least time for \(A\) to intercept \(B\).
    (5)