Questions — Edexcel M4 (159 questions)

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Edexcel M4 2008 June Q1
  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
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
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
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
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
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)
Edexcel M4 2008 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{376d12ab-022c-4070-a1e0-88eacc2fe48e-5_917_814_303_587} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of length \(2 a\) and mass \(k M\) where \(k\) is a constant, is free to rotate in a vertical plane about the fixed point \(A\). One end of a light inextensible string of length \(6 a\) is attached to the end \(B\) of the rod and passes over a small smooth pulley which is fixed at the point \(P\). The line \(A P\) is horizontal and of length \(2 a\). The other end of the string is attached to a particle of mass \(M\) which hangs vertically below the point \(P\), as shown in Figure 3. The angle \(P A B\) is \(2 \theta\), where \(0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\).
  1. Show that the potential energy of the system is $$M g a ( 4 \sin \theta - k \sin 2 \theta ) + \text { constant. }$$ The system has a position of equilibrium when \(\cos \theta = \frac { 3 } { 4 }\).
  2. Find the value of \(k\).
  3. Hence find the value of \(\cos \theta\) at the other position of equilibrium.
  4. Determine the stability of each of the two positions of equilibrium.
Edexcel M4 2009 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f4c33171-597e-4ef3-9f21-3e2271d48f30-02_460_638_230_598} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A fixed smooth plane is inclined to the horizontal at an angle of \(45 ^ { \circ }\). A particle \(P\) is moving horizontally and strikes the plane. Immediately before the impact, \(P\) is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, \(P\) is moving in a direction which makes an angle of \(30 ^ { \circ }\) with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of \(P\) which is lost in the impact.
Edexcel M4 2009 June Q2
2. At time \(t = 0\), a particle \(P\) of mass \(m\) is projected vertically upwards with speed \(\sqrt { \frac { g } { k } }\), where \(k\) is a constant. At time \(t\) the speed of \(P\) is \(v\). The particle \(P\) moves against air resistance whose magnitude is modelled as being \(m k v ^ { 2 }\) when the speed of \(P\) is \(v\). Find, in terms of \(k\), the distance travelled by \(P\) until its speed first becomes half of its initial speed.
(9)
Edexcel M4 2009 June Q3
  1. At noon a motorboat \(P\) is 2 km north-west of another motorboat \(Q\). The motorboat \(P\) is moving due south at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motorboat \(Q\) is pursuing motorboat \(P\) at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and sets a course in order to get as close to motorboat \(P\) as possible.
    1. Find the course set by \(Q\), giving your answer as a bearing to the nearest degree.
    2. Find the shortest distance between \(P\) and \(Q\).
    3. Find the distance travelled by \(Q\) from its position at noon to the point of closest approach.
    \section*{June 2009}
Edexcel M4 2009 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f4c33171-597e-4ef3-9f21-3e2271d48f30-07_479_807_246_571} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.
  1. Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
  2. Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
  3. Determine the stability of the position of equilibrium.
    \section*{June 2009}
Edexcel M4 2009 June Q5
5. Two small smooth spheres \(A\) and \(B\), of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Show that the velocity of \(B\) immediately after the collision is \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the impulse of \(B\) on \(A\) in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to \(\mathbf { i } + \mathbf { j }\).
  3. Find the coefficient of restitution between \(A\) and \(B\).
    \section*{June 2009}
Edexcel M4 2009 June Q6
6. A light elastic spring \(A B\) has natural length \(2 a\) and modulus of elasticity \(2 m n ^ { 2 } a\), where \(n\) is a constant. A particle \(P\) of mass \(m\) is attached to the end \(A\) of the spring. At time \(t = 0\), the spring, with \(P\) attached, lies at rest and unstretched on a smooth horizontal plane. The other end \(B\) of the spring is then pulled along the plane in the direction \(A B\) with constant acceleration \(f\). At time \(t\) the extension of the spring is \(x\).
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
  2. Find \(x\) in terms of \(n , f\) and \(t\). Hence find
  3. the maximum extension of the spring,
  4. the speed of \(P\) when the spring first reaches its maximum extension.
    \section*{June 2009}
Edexcel M4 2010 June Q1
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively]
A man cycles at a constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on level ground and finds that when his velocity is \(u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the velocity of the wind appears to be \(v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(v\) is a positive constant. When the man cycles with velocity \(\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), the velocity of the wind appears to be \(w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(w\) is a positive constant. Find, in terms of \(u\), the true velocity of the wind.
Edexcel M4 2010 June Q2
2. Two smooth uniform spheres \(S\) and \(T\) have equal radii. The mass of \(S\) is 0.3 kg and the mass of \(T\) is 0.6 kg . The spheres are moving on a smooth horizontal plane and collide obliquely. Immediately before the collision the velocity of \(S\) is \(\mathbf { u } _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(T\) is \(\mathbf { u } _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the spheres is 0.5 . Immediately after the collision the velocity of \(S\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(T\) is \(( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Given that when the spheres collide the line joining their centres is parallel to \(\mathbf { i }\),
  1. find
    1. \(\mathbf { u } _ { 1 }\),
    2. \(\mathbf { u } _ { 2 }\). After the collision, \(T\) goes on to collide with a smooth vertical wall which is parallel to \(\mathbf { j }\). Given that the coefficient of restitution between \(T\) and the wall is also 0.5 , find
  2. the angle through which the direction of motion of \(T\) is deflected as a result of the collision with the wall,
  3. the loss in kinetic energy of \(T\) caused by the collision with the wall.
Edexcel M4 2010 June Q3
  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
  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
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
  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
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
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
  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
  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
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
  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.