Questions M4 (327 questions)

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Edexcel M4 Q3
8 marks Challenging +1.8
3. 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 \(\theta , \theta < 45 ^ { \circ }\), 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 \(e = \tan ^ { 2 } \theta\).
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 2008 June Q1
5 marks Standard +0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively.]
A ship \(P\) is moving with velocity ( \(5 \mathbf { i } - 4 \mathbf { j }\) ) \(\mathrm { km } \mathrm { h } ^ { - 1 }\) and a ship \(Q\) is moving with velocity \(( 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Find the direction that ship \(Q\) appears to be moving in, to an observer on ship \(P\), giving your answer as a bearing.
Edexcel M4 2008 June Q2
5 marks Standard +0.3
2. Two small smooth spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(2 m \mathrm {~kg}\) and the mass of \(B\) is \(m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane and they collide. Immediately before the collision the velocity of \(A\) is \(( 2 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the speed of \(B\) immediately after the collision.
(5)
Edexcel M4 2008 June Q3
8 marks Challenging +1.2
3. At time \(t = 0\), a particle of mass \(m\) is projected vertically downwards with speed \(U\) from a point above the ground. At time \(t\) the speed of the particle is \(v\) and the magnitude of the air resistance is modelled as being \(m k v\), where \(k\) is a constant. Given that \(U < \frac { \boldsymbol { g } } { \mathbf { 2 } \boldsymbol { k } }\), find, in terms of \(k , U\) and \(g\), the time taken for the particle to double its speed.
(8)
Edexcel M4 2008 June Q4
8 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{376d12ab-022c-4070-a1e0-88eacc2fe48e-2_451_357_1672_852} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small smooth ball \(B\), moving on a horizontal plane, collides with a fixed vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(2 \theta\), where \(0 ^ { \circ } < \theta < 45 ^ { \circ }\). Immediately after the collision the angle between the direction of motion of \(B\) and the wall is \(\theta\), as shown in Figure 1. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 8 }\), find the value of \(\tan \theta\).
(8)
Edexcel M4 2008 June Q5
15 marks Challenging +1.2
5. A light elastic spring has natural length \(l\) and modulus of elasticity \(m g\). One end of the spring is fixed to a point \(O\) on a rough horizontal table. The other end is attached to a particle \(P\) of mass \(m\) which is at rest on the table with \(O P = l\). At time \(t = 0\) the particle is projected with speed \(\sqrt { } ( g l )\) along the table in the direction \(O P\). At time \(t\) the displacement of \(P\) from its initial position is \(x\) and its speed is \(v\). The motion of \(P\) is subject to air resistance of magnitude \(2 m v \omega\), where \(\omega = \sqrt { \frac { g } { l } }\). The coefficient of friction between \(P\) and the table is 0.5 .
  1. Show that, until \(P\) first comes to rest, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + \omega ^ { 2 } x = - 0.5 g$$
  2. Find \(x\) in terms of \(t , l\) and \(\omega\).
  3. Hence find, in terms of \(\omega\), the time taken for \(P\) to first come to instantaneous rest.
    (3)
Edexcel M4 2008 June Q6
16 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{376d12ab-022c-4070-a1e0-88eacc2fe48e-4_448_803_242_630} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A river is 30 m wide and flows between two straight parallel banks. At each point of the river, the direction of flow is parallel to the banks. At time \(t = 0\), a boat leaves a point \(O\) on one bank and moves in a straight line across the river to a point \(P\) on the opposite bank. Its path \(O P\) is perpendicular to both banks and \(O P = 30 \mathrm {~m}\), as shown in Figure 2. The speed of flow of the river, \(r \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at a point on \(O P\) which is at a distance \(x \mathrm {~m}\) from \(O\), is modelled as $$r = \frac { 1 } { 10 } x , \quad 0 \leq x \leq 30$$ The speed of the boat relative to the water is constant at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds the boat is at a distance \(x \mathrm {~m}\) from \(O\) and is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(O P\).
  1. Show that $$100 v ^ { 2 } = 2500 - x ^ { 2 }$$
  2. Hence show that $$\frac { \mathbf { d } ^ { 2 } x } { \mathbf { d } t ^ { 2 } } + \frac { x } { 100 } = 0$$
  3. Find the total time taken for the boat to cross the river from \(O\) to \(P\).
    (9)