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Edexcel M4 2016 June Q4
12 marks Standard +0.3
4. A particle \(P\) of mass 9 kg moves along the horizontal positive \(x\)-axis under the action of a force directed towards the origin. At time \(t\) seconds, the displacement of \(P\) from \(O\) is \(x\) metres, \(P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the force has magnitude \(16 x\) newtons. The particle \(P\) is also subject to a resistive force of magnitude \(24 v\) newtons.
  1. Show that the equation of motion of \(P\) is $$9 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 24 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 16 x = 0$$ It is given that the general solution of this differential equation is $$x = \mathrm { e } ^ { - \frac { 4 } { 3 } t } ( A t + B )$$ where \(A\) and \(B\) are arbitrary constants.
    When \(t = \frac { 3 } { 4 } , P\) is travelling towards \(O\) with its maximum speed of \(8 \mathrm { e } ^ { - 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(x = d\).
  2. Find the value of \(d\).
  3. Find the value of \(x\) when \(t = 0\)
Edexcel M4 2016 June Q5
17 marks Challenging +1.8
5. A toy car of mass 0.5 kg is attached to one end \(A\) of a light elastic string \(A B\), of natural length 1.5 m and modulus of elasticity 27 N . Initially the car is at rest on a smooth horizontal floor and the string lies in a straight line with \(A B = 1.5 \mathrm {~m}\). The end \(B\) is moved in a straight horizontal line directly away from the car, with constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds after \(B\) starts to move, the extension of the string is \(x\) metres and the car has moved a distance \(y\) metres. The effect of air resistance on the car can be ignored. By modelling the car as a particle, show that, while the string remains taut,
    1. \(x + y = u t\)
    2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 36 x = 0\)
  2. Hence show that the string becomes slack when \(t = \frac { \pi } { 6 }\)
  3. Find, in terms of \(u\), the speed of the car when \(t = \frac { \pi } { 12 }\)
  4. Find, in terms of \(u\), the distance the car has travelled when it first reaches end \(B\) of the string.
Edexcel M4 2016 June Q6
16 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b01b3a41-3ed1-4104-b20d-4cfb845df4a1-11_664_786_221_587} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a uniform rod \(A B\), of length \(2 l\) and mass \(4 m\). A particle of mass \(2 m\) is attached to the rod at \(B\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic spring, of natural length \(2 l\) and modulus of elasticity \(k m g\), where \(k > 4\), is attached to the rod at \(B\). The other end of the spring is attached to a fixed point \(C\) which is vertically above \(A\), where \(A C = 2 l\). The angle \(B A C\) is \(2 \theta\), where \(\frac { \pi } { 6 } < \theta \leqslant \frac { \pi } { 2 }\)
  1. Show that the potential energy of the system is $$4 m g l \left\{ ( k - 4 ) \sin ^ { 2 } \theta - k \sin \theta \right\} + \text { constant }$$ Given that there is a position of equilibrium with \(\theta \neq \frac { \pi } { 2 }\)
  2. show that \(k > 8\) Given that \(k = 10\)
  3. determine the stability of this position of equilibrium.
Edexcel M4 2017 June Q1
8 marks Standard +0.8
  1. \hspace{0pt} [In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]
A ship \(A\) has constant velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) and a ship \(B\) has constant velocity \(( - \mathbf { i } + 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At noon, the position vectors of the ships \(A\) and \(B\) with respect to a fixed origin \(O\) are \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { km }\) and \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) respectively. Find
  1. the time at which the two ships are closest together,
  2. the length of time for which ship \(A\) is within 2 km of ship \(B\).
Edexcel M4 2017 June Q2
12 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23e6a9ae-bf00-45a3-b462-e18760d9af45-04_912_988_260_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(3 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). 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 two spheres is \(e\).
The kinetic energy of sphere \(B\) immediately after the collision is \(85 \%\) of its kinetic energy immediately before the collision. Find
  1. the velocity of each sphere immediately after the collision,
  2. the value of \(e\).
Edexcel M4 2017 June Q3
12 marks Challenging +1.2
3. A cyclist and her bicycle have a combined mass of 75 kg . The cyclist travels along a straight horizontal road. The cyclist produces a constant driving force of magnitude 150 N . At time \(t\) seconds, the speed of the cyclist is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v < \sqrt { 50 }\). As the cyclist moves, the total resistance to motion of the cyclist and her bicycle has magnitude \(3 v ^ { 2 }\) newtons. The cyclist starts from rest. At time \(t\) seconds, she has travelled a distance \(x\) metres from her starting point. Find
  1. \(v\) in terms of \(x\),
  2. \(t\) in terms of \(v\).
Edexcel M4 2017 June Q4
8 marks Standard +0.8
4. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] A line of greatest slope of a fixed smooth plane is parallel to the vector \(( - 4 \mathbf { i } - 3 \mathbf { j } )\). A particle \(P\) falls vertically and strikes the plane. Immediately before the impact, \(P\) has velocity \(- 7 \mathbf { j } \mathrm {~ms} ^ { - 1 }\). Immediately after the impact, \(P\) has velocity \(( - a \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(a\) is a positive constant.
  1. Show that \(a = 6\)
  2. Find the coefficient of restitution between \(P\) and the plane.
Edexcel M4 2017 June Q5
9 marks Challenging +1.2
5. A cyclist riding due north at a steady speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) notices that the wind appears to come from the north-west. At the same time, another cyclist, moving on a bearing of \(120 ^ { \circ }\) and also riding at a steady speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), notices that the wind appears to come from due south. The velocity of the wind is assumed to be constant. Find
  1. the wind speed,
  2. the direction from which the wind is blowing, giving your answer as a bearing.
Edexcel M4 2017 June Q6
13 marks Standard +0.8
6. A particle \(P\) of mass 0.2 kg is suspended from a fixed point by a light elastic spring. The spring has natural length 0.8 m and modulus of elasticity 7 N . At time \(t = 0\) the particle is released from rest from a point 0.2 metres vertically below its equilibrium position. The motion of \(P\) is resisted by a force of magnitude \(2 v\) newtons, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\). At time \(t\) seconds, \(P\) is \(x\) metres below its equilibrium position.
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 10 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 43.75 x = 0\)
  2. Find \(x\) in terms of \(t\).
  3. Find the value of \(t\) when \(P\) first comes to instantaneous rest.
Edexcel M4 2017 June Q7
13 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23e6a9ae-bf00-45a3-b462-e18760d9af45-24_655_890_239_529} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows four uniform rods, each of mass \(m\) and length \(2 a\). The rods are freely hinged at their ends to form a rhombus \(A B C D\). Point \(A\) is attached to a fixed point on a ceiling and the rhombus hangs freely with \(C\) vertically below \(A\). A light elastic spring of natural length \(2 a\) and modulus of elasticity \(7 m g\) connects the points \(A\) and \(C\). A particle of mass \(3 m\) is attached to point \(C\).
  1. Show that, when \(A D\) is at an angle \(\theta\) to the downward vertical, the potential energy \(V\) of the system is given by $$V = 28 m g a \cos ^ { 2 } \theta - 48 m g a \cos \theta + \text { constant }$$ Given that \(\theta > 0\)
  2. find the value of \(\theta\) for which the system is in equilibrium,
  3. determine the stability of this position of equilibrium.
Edexcel M4 2018 June Q1
11 marks Challenging +1.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0f141c7-ecd0-4f62-bfad-76c81c2d6396-02_538_881_278_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\) has mass \(m\) and length 4a. The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string, of natural length \(a\) and modulus \(\frac { 1 } { 4 } m g\), is attached to the end \(B\) of the rod. The other end of the string is attached to a small light smooth ring \(R\). The ring can move freely on a smooth horizontal wire which is fixed at a height \(a\) above \(A\), and in a vertical plane through \(A\). The angle between the rod and the horizontal is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1. Given that the elastic string is vertical,
  1. show that the potential energy of the system is $$2 m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) + \text { constant }$$
  2. Show that when \(\theta = \frac { \pi } { 6 }\) the rod is in stable equilibrium.
Edexcel M4 2018 June Q2
8 marks Standard +0.8
2. A small ball \(B\), moving on a smooth horizontal plane, collides with a fixed smooth vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(\alpha\). The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 4 }\). The kinetic energy of \(B\) immediately after the collision is \(60 \%\) of its kinetic energy immediately before the collision. Find, in degrees, the size of angle \(\alpha\).
Edexcel M4 2018 June Q3
7 marks Challenging +1.2
3. When a man walks due West at a constant speed of \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the wind appears to be blowing from due South. When he runs due North at a constant speed of \(8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the speed of the wind appears to be \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
The velocity of the wind relative to the Earth is constant with magnitude \(w \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
Find the two possible values of \(w\).
Edexcel M4 2018 June Q4
11 marks Challenging +1.2
4. A particle \(P\) of mass 0.5 kg moves in a horizontal straight line. At time \(t\) seconds \(( t \geqslant 0 )\), the displacement of \(P\) from a fixed point \(O\) of the line is \(x\) metres, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(P\) is moving in the direction of \(x\) increasing. A force of magnitude \(k x\) newtons acts on \(P\) in the direction \(P O\). The motion of \(P\) is also subject to a resistance of magnitude \(\lambda v\) newtons. Given that $$x = ( 1.5 + 10 t ) \mathrm { e } ^ { - 4 t }$$ find
  1. the value of \(k\) and the value of \(\lambda\),
  2. the distance from \(P\) to \(O\) when \(P\) is instantaneously at rest.
Edexcel M4 2018 June Q5
11 marks Standard +0.8
5. A horizontal square field, \(P Q R S\), has sides of length 75 m . Ali is at corner \(P\) of the field and Beth is at corner \(Q\) of the field. Ali starts to walk in a straight line along the diagonal of the field from \(P\) to \(R\) at a constant speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Beth sees Ali start to walk, waits 10 seconds, and then walks from \(Q\) to intercept Ali. Beth walks in a straight line at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the time from the instant Beth leaves \(Q\) until the instant that she intercepts Ali,
  2. the direction Beth should take.
Edexcel M4 2018 June Q6
14 marks Challenging +1.2
6. A particle of mass \(m\) is projected vertically upwards in a resisting medium. As the particle moves upwards, the speed \(v\) of the particle is given by $$v ^ { 2 } = k g \left( 5 \mathrm { e } ^ { - \frac { x } { 2 k } } - 4 \right)$$ where \(x\) is the distance of the particle above the point of projection and \(k\) is a positive constant.
  1. Show that the magnitude of the resistance to the motion of the particle is \(\frac { m v ^ { 2 } } { 4 k }\).
    (4)
  2. Find, in terms of \(k\), the greatest height reached by the particle above the point of projection.
  3. Show that the time taken by the particle to reach its greatest height above the point of projection is \(\sqrt { \frac { 4 k } { g } } \arctan \left( \frac { 1 } { 2 } \right)\)
Edexcel M4 2018 June Q7
13 marks Standard +0.8
7. Two smooth uniform spheres \(A\) and \(B\), of mass 2 kg and 3 kg respectively, and of equal radius, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Show that, at the instant when \(A\) and \(B\) collide, their line of centres is parallel to \(- \mathbf { i } + \mathbf { j }\).
  2. Find the velocity of \(B\) immediately after the collision.
  3. Find the coefficient of restitution between \(A\) and \(B\).
Edexcel M4 Q1
6 marks Standard +0.3
  1. A smooth sphere \(S\) is moving on a smooth horizontal plane with speed \(u\) when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of \(S\) makes an angle of \(30 ^ { \circ }\) with the wall. The coefficient of restitution between \(S\) and the wall is \(\frac { 1 } { 3 }\).
Find the speed of \(S\) immediately after the collision.
Edexcel M4 Q2
8 marks Challenging +1.2
2. A car of mass 1000 kg , moving along a straight horizontal road, is driven by an engine which produces a constant power of 12 kW . The only resistance to the motion of the car is air resistance of magnitude \(10 v ^ { 2 } \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car. Find the distance travelled by the car as its speed increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8 marks)
Edexcel M4 Q3
10 marks Challenging +1.2
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-3_469_1163_1217_443}
\end{figure} A smooth uniform sphere \(A\), moving on a smooth horizontal table, collides with a second identical sphere \(B\) which is at rest on the table. When the spheres collide the line joining their centres makes an angle of \(30 ^ { \circ }\) with the direction of motion of \(A\), as shown in Fig. 1. The coefficient of restitution between the spheres is \(e\). The direction of motion of \(A\) is deflected through an angle \(\theta\) by the collision. Show that \(\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }\).
(10 marks)
Edexcel M4 Q4
10 marks Standard +0.8
4. A body falls vertically from rest and is subject to air resistance of a magnitude which is proportional to its speed. Given that its terminal speed is \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time it takes for the body to attain a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(10 marks)
Edexcel M4 Q5
12 marks Standard +0.8
5. A particle \(P\) of mass \(m\) is fixed to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m a n ^ { 2 }\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at a point which is a distance \(2 a\) vertically below \(O\). The air resistance is modelled as having magnitude \(2 m n v\), where \(v\) is the speed of \(P\).
  1. Show that, when the extension of the string is \(x\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 n \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 n ^ { 2 } x = g$$
  2. Find \(x\) in terms of \(t\).
Edexcel M4 Q6
12 marks Challenging +1.8
6. Two particles \(P\) and \(Q\) have constant velocity vectors \(\mathbf { v } _ { P }\) and \(\mathbf { v } _ { Q }\) respectively. The magnitude of the velocity of \(P\) relative to \(Q\) is equal to the speed of \(P\). If the direction of motion of one of the particles is reversed, the magnitude of the velocity of \(P\) relative to \(Q\) is doubled. Find
  1. the ratio of the speeds of \(P\) and \(Q\),
  2. the cosine of the angle between the directions of motion of \(P\) and \(Q\).
Edexcel M4 Q7
17 marks Challenging +1.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-5_955_855_349_573}
\end{figure} A smooth wire \(A B\), in the shape of a circle of radius \(r\), is fixed in a vertical plane with \(A B\) vertical. A small smooth ring \(R\) of mass \(m\) is threaded on the wire and is connected by a light inextensible string to a particle \(P\) of mass \(m\). The length of the string is greater than the diameter of the circle. The string passes over a small smooth pulley which is fixed at the highest point \(A\) of the wire and angle \(R \hat { A } P = \theta\), as shown in Fig. 2.
  1. Show that the potential energy of the system is given by $$2 m g r \left( \cos \theta - \cos ^ { 2 } \theta \right) + \text { constant. }$$
  2. Hence determine the values of \(\theta , \theta \geq 0\), for which the system is in equilibrium. (6 marks)
  3. Determine the stability of each position of equilibrium. END
Edexcel M4 Specimen Q1
6 marks Moderate -0.3
  1. A particle \(P\) of mass 2 kg moves in a straight line along a smooth horizontal plane. The only horizontal force acting on \(P\) is a resistance of magnitude \(4 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is its speed. At time \(t = 0 \mathrm {~s} , P\) has a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(v\) in terms of \(t\).
    (6)
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-2_470_979_657_591}
\end{figure} A girl swims in still water at \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She swims across a river which is 336 m wide and is flowing at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She sets off from a point \(A\) on one bank and lands at a point \(B\), which is directly opposite \(A\), on the other bank as shown in Fig. 1. Find
  1. the direction, relative to the earth, in which she swims,
  2. the time that she takes to cross the river.