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Edexcel M4 2010 June Q3
10 marks Standard +0.8
  1. At 12 noon, \(\operatorname { ship } A\) is 8 km due west of \(\operatorname { ship } B\). Ship \(A\) is moving due north at a constant speed of \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Ship \(B\) is moving at a constant speed of \(6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing so that it passes as close to \(A\) as possible.
    1. Find the bearing on which ship \(B\) moves.
    2. Find the shortest distance between the two ships.
    3. Find the time when the two ships are closest.
Edexcel M4 2010 June Q4
12 marks Challenging +1.8
  1. A particle of mass \(m\) is projected vertically upwards, at time \(t = 0\), with speed \(U\). The particle is subject to air resistance of magnitude \(\frac { m g v ^ { 2 } } { k ^ { 2 } }\), where \(v\) is the speed of the particle at time \(t\) and \(k\) is a positive constant.
    1. Show that the particle reaches its greatest height above the point of projection at time
    $$\frac { k } { g } \tan ^ { - 1 } \left( \frac { U } { k } \right)$$
  2. Find the greatest height above the point of projection attained by the particle.
Edexcel M4 2010 June Q5
15 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{60202547-5d12-405f-bc83-2907419ec354-09_413_1212_262_365} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The end \(A\) of a uniform rod \(A B\), of length \(2 a\) and mass \(4 m\), is smoothly hinged to a fixed point. The end \(B\) is attached to one end of a light inextensible string which passes over a small smooth pulley, fixed at the same level as \(A\). The distance from \(A\) to the pulley is \(4 a\). The other end of the string carries a particle of mass \(m\) which hangs freely, vertically below the pulley, with the string taut. The angle between the rod and the downward vertical is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1.
  1. Show that the potential energy of the system is $$2 m g a ( \sqrt { } ( 5 - 4 \sin \theta ) - 2 \cos \theta ) + \text { constant }$$
  2. Hence, or otherwise, show that any value of \(\theta\) which corresponds to a position of equilibrium of the system satisfies the equation $$4 \sin ^ { 3 } \theta - 6 \sin ^ { 2 } \theta + 1 = 0 .$$
  3. Given that \(\theta = \frac { \pi } { 6 }\) corresponds to a position of equilibrium, determine its stability. \section*{L \(\_\_\_\_\)}
Edexcel M4 2010 June Q6
17 marks Challenging +1.2
  1. Two points \(A\) and \(B\) lie on a smooth horizontal table with \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(2 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) is on the table at rest and in equilibrium.
    1. Show that \(A P = \frac { 5 a } { 3 }\).
    The particle \(P\) is now moved along the table from its equilibrium position through a distance \(0.5 a\) towards \(B\) and released from rest at time \(t = 0\). At time \(t , P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. There is a resistance to motion of magnitude \(4 m \omega v\) where \(\omega = \sqrt { } \left( \frac { g } { a } \right)\).
  2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 3 \omega ^ { 2 } x = 0\).
  3. Find the velocity, \(\frac { \mathrm { d } x } { \mathrm {~d} t }\), of \(P\) in terms of \(a , \omega\) and \(t\).
Edexcel M4 2011 June Q1
10 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2b891a9c-3abe-4e88-ba94-b6abcb37b4c3-02_794_1022_214_488} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When the spheres collide, the line joining their centres is parallel to \(\mathbf { j }\), as shown in Figure 1. The coefficient of restitution between the spheres is \(\frac { 3 } { 7 }\). Find, in terms of \(m\), the total kinetic energy lost in the collision.
Edexcel M4 2011 June Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2b891a9c-3abe-4e88-ba94-b6abcb37b4c3-04_682_853_283_543} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 represents part of the smooth rectangular floor of a sports hall. A ball is at \(B\), 4 m from one wall of the hall and 5 m from an adjacent wall. These two walls are smooth and meet at the corner \(C\). The ball is kicked so that it travels along the floor, bounces off the first wall at the point \(X\) and hits the second wall at the point \(Y\). The point \(Y\) is 7.5 m from the corner \(C\).
The coefficient of restitution between the ball and the first wall is \(\frac { 3 } { 4 }\).
Modelling the ball as a particle, find the distance \(C X\).
Edexcel M4 2011 June Q3
11 marks Standard +0.8
  1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]
A coastguard patrol boat \(C\) is moving with constant velocity \(( 8 \mathbf { i } + u \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Another ship \(S\) is moving with constant velocity \(( 12 \mathbf { i } + 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Find, in terms of \(u\), the velocity of \(C\) relative to \(S\). At noon, \(S\) is 10 km due west of \(C\).
    If \(C\) is to intercept \(S\),
    1. find the value of \(u\).
    2. Using this value of \(u\), find the time at which \(C\) would intercept \(S\). If instead, at noon, \(C\) is moving with velocity \(( 8 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and continues at this constant velocity,
  3. find the distance of closest approach of \(C\) to \(S\).
Edexcel M4 2011 June Q4
7 marks Challenging +1.2
  1. A hiker walking due east at a steady speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) notices that the wind appears to come from a direction with bearing 050. At the same time, another hiker moving on a bearing of 320, and also walking at \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), notices that the wind appears to come from due north.
Find
  1. the direction from which the wind is blowing,
  2. the wind speed.
Edexcel M4 2011 June Q5
11 marks Challenging +1.2
5. A particle \(Q\) of mass 6 kg is moving along the \(x\)-axis. At time \(t\) seconds the displacement of \(Q\) from the origin \(O\) is \(x\) metres and the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle moves under the action of a retarding force of magnitude ( \(a + b v ^ { 2 }\) ) N, where \(a\) and \(b\) are positive constants. At time \(t = 0 , Q\) is at \(O\) and moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. The particle \(Q\) comes to instantaneous rest at the point \(X\).
  1. Show that the distance \(O X\) is $$\frac { 3 } { b } \ln \left( 1 + \frac { b U ^ { 2 } } { a } \right) \mathrm { m }$$ Given that \(a = 12\) and \(b = 3\),
  2. find, in terms of \(U\), the time taken to move from \(O\) to \(X\).
Edexcel M4 2011 June Q6
13 marks Standard +0.3
  1. A particle \(P\) of mass 4 kg moves along a horizontal straight line under the action of a force directed towards a fixed point \(O\) on the line. At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and the force towards \(O\) has magnitude \(9 x\) newtons. The particle \(P\) is also subject to air resistance, which has magnitude \(12 v\) newtons when \(P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that the equation of motion of \(P\) is
    $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = 0$$ It is given that the solution of this differential equation is of the form $$x = \mathrm { e } ^ { - \lambda t } ( A t + B )$$ When \(t = 0\) the particle is released from rest at the point \(R\), where \(O R = 4 \mathrm {~m}\). Find,
  2. the values of the constants \(\lambda , A\) and \(B\),
  3. the greatest speed of \(P\) in the subsequent motion.
Edexcel M4 2011 June Q7
14 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2b891a9c-3abe-4e88-ba94-b6abcb37b4c3-13_451_1077_315_370} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The rod \(A B\) has length \(2 a\) and mass \(4 m\), and the rod \(B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).
  1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
  2. Find the value of \(\theta\) for which the system is in equilibrium.
  3. Determine the stability of this position of equilibrium.
Edexcel M4 2012 June Q1
13 marks Challenging +1.2
  1. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).
    1. Find, in terms of \(V\), the speed of \(S\)
      1. immediately before the collision,
      2. immediately after the collision.
    2. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
Edexcel M4 2012 June Q2
13 marks Standard +0.8
  1. A \(\operatorname { ship } A\) is moving at a constant speed of \(8 \mathrm {~km} \mathrm {~h} \mathrm {~h} ^ { - 1 }\) on a bearing of \(150 ^ { \circ }\). At noon a second ship \(B\) is 6 km from \(A\), on a bearing of \(210 ^ { \circ }\). Ship \(B\) is moving due east at a constant speed. At a later time, \(B\) is \(2 \sqrt { 3 } \mathrm {~km}\) due south of \(A\).
Find
  1. the time at which \(B\) will be due east of \(A\),
  2. the distance between the ships at that time.
Edexcel M4 2012 June Q3
16 marks Challenging +1.2
  1. Two particles, of masses \(m\) and \(2 m\), are connected to the ends of a long light inextensible string. The string passes over a small smooth fixed pulley and hangs vertically on either side. The particles are released from rest with the string taut. Each particle is subject to air resistance of magnitude \(k v ^ { 2 }\), where \(v\) is the speed of each particle after it has moved a distance \(x\) from rest and \(k\) is a positive constant.
    1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( v ^ { 2 } \right) + \frac { 4 k } { 3 m } v ^ { 2 } = \frac { 2 g } { 3 }\)
    2. Find \(v ^ { 2 }\) in terms of \(x\).
    3. Deduce that the tension in the string, \(T\), satisfies
    $$\frac { 4 m g } { 3 } \leqslant T < \frac { 3 m g } { 2 }$$
Edexcel M4 2012 June Q4
10 marks Challenging +1.2
4. A rescue boat, whose maximum speed is \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), receives a signal which indicates that a yacht is in distress near a fixed point \(P\). The rescue boat is 15 km south-west of \(P\). There is a constant current of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) flowing uniformly from west to east. The rescue boat sets the course needed to get to \(P\) as quickly as possible. Find
  1. the course the rescue boat sets,
  2. the time, to the nearest minute, to get to \(P\). When the rescue boat arrives at \(P\), the yacht is just visible 4 km due north of \(P\) and is drifting with the current. Find
  3. the course that the rescue boat should set to get to the yacht as quickly as possible,
  4. the time taken by the rescue boat to reach the yacht from \(P\).
Edexcel M4 2012 June Q5
12 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{07536810-a589-4820-a330-78c35022eb68-10_977_1224_205_360} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\), of length \(4 a\) and weight \(W\), is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through the point \(C\) of the rod, where \(A C = 3 a\). One end of a light inextensible string of length \(L\), where \(L > 10 a\), is attached to the end \(A\) of the rod and passes over a small smooth fixed peg at \(P\) and another small smooth fixed peg at \(Q\). The point \(Q\) lies in the same vertical plane as \(P , A\) and \(B\). The point \(P\) is at a distance \(3 a\) vertically above \(C\) and \(P Q\) is horizontal with \(P Q = 4 a\). A particle of weight \(\frac { 1 } { 2 } W\) is attached to the other end of the string and hangs vertically below \(Q\). The rod is inclined at an angle \(2 \theta\) to the vertical, where \(- \pi < 2 \theta < \pi\), as shown in Figure 1.
  1. Show that the potential energy of the system is $$W a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
  2. Find the positions of equilibrium and determine their stability.
Edexcel M4 2012 June Q6
11 marks Challenging +1.2
6. Two points \(A\) and \(B\) are in a vertical line, with \(A\) above \(B\) and \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) hangs at rest in equilibrium.
  1. Show that \(A P = \frac { 7 a } { 4 }\) The particle \(P\) is now pulled down vertically from its equilibrium position towards \(B\) and at time \(t = 0\) it is released from rest. At time \(t\), the particle \(P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. The particle \(P\) is subject to air resistance of magnitude \(m k v\), where \(k\) is a positive constant.
  2. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
  3. Find the range of values of \(k\) which would result in the motion of \(P\) being a damped oscillation.
Edexcel M4 2013 June Q1
5 marks Moderate -0.5
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.]
Boat \(A\) is moving with velocity ( \(3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and boat \(B\) is moving with velocity \(( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Find
  1. the magnitude of the velocity of \(A\) relative to \(B\),
  2. the direction of the velocity of \(A\) relative to \(B\), giving your answer as a bearing.
Edexcel M4 2013 June Q2
6 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a3ae838-b58e-4957-8d98-f7d8a65df99a-03_604_741_123_605} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A smooth fixed plane is inclined at an angle \(\alpha\) to the horizontal. A smooth ball \(B\) falls vertically and hits the plane. Immediately before the impact the speed of \(B\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Immediately after the impact the direction of motion of \(B\) is horizontal. The coefficient of restitution between \(B\) and the plane is \(\frac { 1 } { 3 }\). Find the size of angle \(\alpha\).
Edexcel M4 2013 June Q3
9 marks Challenging +1.2
  1. A smooth uniform sphere \(A\), of mass \(5 m\) and radius \(r\), is at rest on a smooth horizontal plane. A second smooth uniform sphere \(B\), of mass \(3 m\) and radius \(r\), is moving in a straight line on the plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes \(A\). Immediately before the impact the direction of motion of \(B\) makes an angle of \(60 ^ { \circ }\) with the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(30 ^ { \circ }\) by the impact.
Find
  1. the speed of \(B\) immediately after the impact,
  2. the coefficient of restitution between the spheres.
Edexcel M4 2013 June Q4
10 marks Challenging +1.2
  1. At 10 a.m. two walkers \(A\) and \(B\) are 4 km apart with \(A\) due north of \(B\). Walker \(A\) is moving due east at a constant speed of \(6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Walker \(B\) is moving with constant speed \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and walks in the straight line which allows him to pass as close as possible to \(A\).
Find
  1. the direction of motion of \(B\), giving your answer as a bearing,
  2. the least distance between \(A\) and \(B\),
  3. the time when the distance between \(A\) and \(B\) is least.
Edexcel M4 2013 June Q5
12 marks Challenging +1.2
5. A van of mass 1200 kg travels along a straight horizontal road against a resistance to motion which is proportional to the speed of the van. The engine of the van is working at a constant rate of 40 kW . The van starts from rest at time \(t = 0\). At time \(t\) seconds, the speed of the van is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed of the van is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that $$75 v \frac { \mathrm {~d} v } { \mathrm {~d} t } = 2500 - v ^ { 2 }$$
  2. Find \(v\) in terms of \(t\).
Edexcel M4 2013 June Q6
16 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a3ae838-b58e-4957-8d98-f7d8a65df99a-11_573_679_248_685} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod \(A B\) has mass \(4 m\) and length \(4 l\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). A particle of mass \(k m\), where \(k < 7\), is attached to the rod at \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity 4 mg , is attached to the point \(D\) of the rod, where \(A D = 3 l\). The other end of the string is attached to a fixed point \(E\) which is vertically above \(A\), where \(A E = 3 l\), as shown in Figure 2. The angle between the rod and the upward vertical is \(2 \theta\), where \(\arcsin \left( \frac { 1 } { 6 } \right) < \theta \leqslant \frac { \pi } { 2 }\).
  1. Show that, while the string is stretched, the potential energy of the system is $$8 m g l \left\{ ( 7 - k ) \sin ^ { 2 } \theta - 3 \sin \theta \right\} + \text { constant }$$ There is a position of equilibrium with \(\theta \leqslant \frac { \pi } { 6 }\).
  2. Show that \(k \leqslant 4\) Given that \(k = 4\),
  3. show that this position of equilibrium is stable.
Edexcel M4 2013 June Q7
17 marks Standard +0.8
7. A particle \(P\) of mass 0.5 kg is attached to the end \(A\) of a light elastic spring \(A B\), of natural length 0.6 m and modulus of elasticity 2.7 N . At time \(t = 0\) the end \(B\) of the spring is held at rest and \(P\) hangs at rest at the point \(C\) which is vertically below \(B\). The end \(B\) is then moved along the line of the spring so that, at time \(t\) seconds, the downwards displacement of \(B\) from its initial position is \(4 \sin 2 t\) metres. At time \(t\) seconds, the extension of the spring is \(x\) metres and the displacement of \(P\) below \(C\) is \(y\) metres.
  1. Show that $$y + \frac { 49 } { 45 } = x + 4 \sin 2 t$$
  2. Hence show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 9 y = 36 \sin 2 t$$ Given that \(y = \frac { 36 } { 5 } \sin 2 t\) is a particular integral of this differential equation,
  3. find \(y\) in terms of \(t\),
  4. find the speed of \(P\) when \(t = \frac { 1 } { 3 } \pi\).
Edexcel M4 2013 June Q1
13 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg falls vertically from rest. After \(t\) seconds it has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude 1.5 v newtons acts on \(P\) as it falls.
    1. Show that \(3 v = 9.8 \left( 1 - \mathrm { e } ^ { - 3 t } \right)\).
    2. Find the distance that \(P\) falls in the first two seconds of its motion.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a4cdf2b0-8dd0-4c12-9259-95b78875c6cb-03_410_919_219_532} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A river is 50 m wide and flows between two straight parallel banks. The river flows with a uniform speed of \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. The points \(A\) and \(B\) are on opposite banks of the river and \(A B\) 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 \(\frac { 10 } { 9 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Keith sets out from \(A\) and crosses the river in the least possible time, reaching the opposite bank at the point \(C\). Find
  2. the time taken by Keith to reach \(C\),
  3. the distance \(B C\). Ian sets out from \(A\) and swims in a straight line so as to land on the opposite bank at \(B\).
  4. Find the time taken by Ian to reach \(B\).