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Edexcel M3 Q4
9 marks Standard +0.3
4. A particle moves with simple harmonic motion along a straight line. When the particle is 3 cm from its centre of motion it has a speed of \(8 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) and an acceleration of magnitude \(12 \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).
  1. Show that the period of the motion is \(\pi\) seconds.
  2. Find the amplitude of the motion.
  3. Hence, find the greatest speed of the particle.
Edexcel M3 Q5
10 marks Standard +0.3
5. A physics student is set the task of finding the mass of an object without using a set of scales. She decides to use a light elastic string of natural length 2 m and modulus of elasticity 280 N attached to two points \(A\) and \(B\) which are on the same horizontal level and 2.4 m apart. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-3_307_1072_993_438} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} She attaches the object to the midpoint of the string so that it hangs in equilibrium 0.35 m below \(A B\) as shown in Figure 2.
  1. Explain why it is reasonable to assume that the tensions in each half of the string are equal.
  2. Find the mass of the object.
  3. Find the elastic potential energy of the string when the object is suspended from it.
Edexcel M3 Q6
13 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-4_455_540_201_660} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows part of the curve \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
  2. Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
    (3 marks)
Edexcel M3 Q7
20 marks Standard +0.8
7. A particle of mass 0.5 kg is hanging vertically at one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point. The particle is given an initial horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\).
  1. Show that the particle will perform complete circles if \(u \geq \sqrt { 3 g }\). Given that \(u = 5\),
  2. find, correct to the nearest degree, the angle through which the string turns before it becomes slack,
  3. find, correct to the nearest centimetre, the greatest height the particle reaches above its position when the string becomes slack.
Edexcel M3 Q1
8 marks Moderate -0.3
  1. A particle \(P\) of mass 1.5 kg moves from rest at the origin such that at time \(t\) seconds it is subject to a single force of magnitude \(( 4 t + 3 ) \mathrm { N }\) in the direction of the positive \(x\)-axis.
    1. Find the magnitude of the impulse exerted by the force during the interval \(1 \leq t \leq 4\).
    Given that at time \(T\) seconds, \(P\) has a speed of \(22 \mathrm {~ms} ^ { - 1 }\),
  2. find the value of \(T\) correct to 3 significant figures.
Edexcel M3 Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-2_469_465_776_680} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle \(P\) of mass 0.5 kg is at rest at the highest point \(A\) of a smooth sphere, centre \(O\), of radius 1.25 m which is fixed to a horizontal surface. When \(P\) is slightly disturbed it slides along the surface of the sphere. Whilst \(P\) is in contact with the sphere it has speed \(v \mathrm {~ms} ^ { - 1 }\) when \(\angle A O P = \theta\) as shown in Figure 1.
  1. Show that \(v ^ { 2 } = 24.5 ( 1 - \cos \theta )\).
  2. Find the value of \(\cos \theta\) when \(P\) leaves the surface of the sphere.
Edexcel M3 Q3
12 marks Standard +0.8
3. A car starts from rest at the point \(O\) and moves along a straight line. The car accelerates to a maximum velocity, \(V \mathrm {~ms} ^ { - 1 }\), before decelerating and coming to rest again at the point \(A\). The acceleration of the car during this journey, \(a \mathrm {~ms} ^ { - 2 }\), is modelled by the formula $$a = \frac { 500 - k x } { 150 }$$ where \(x\) is the distance in metres of the car from \(O\).
Using this model and given that the car is travelling at \(16 \mathrm {~ms} ^ { - 1 }\) when it is 40 m from \(O\),
  1. find \(k\),
  2. show that \(V = 41\), correct to 2 significant figures,
  3. find the distance \(O A\).
Edexcel M3 Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-3_316_536_1087_639} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\). \(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),
  1. show that the work done against friction is 10.78 J ,
  2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
  3. find \(\lambda\).
Edexcel M3 Q5
16 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-4_693_554_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A flask is modelled as a uniform solid formed by removing a cylinder of radius \(r\) and height \(h\) from a cylinder of radius \(\frac { 4 } { 3 } r\) and height \(\frac { 3 } { 2 } h\) with the same axis of symmetry and a common plane as shown in Figure 3.
  1. Show that the centre of mass of the flask is a distance of \(\frac { 9 } { 10 } h\) from the open end of the flask. The flask is made from a material of density \(\rho\) and is filled to the level of the open plane face with a liquid of density \(k \rho\). Given that the centre of mass of the flask and liquid together is a distance of \(\frac { 15 } { 22 } h\) from the open end of the flask,
  2. find the value of \(k\).
  3. Explain why it may be advantageous to make the base of the flask from a more dense material.
    (2 marks)
Edexcel M3 Q6
19 marks Standard +0.8
6. A particle \(P\) of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points \(A\) and \(B\) on a smooth horizontal table. When \(P\) is 3 m from \(O\), the centre of the oscillations, its speed is \(6 \mathrm {~ms} ^ { - 1 }\). When \(P\) is 2.25 m from \(O\), its speed is \(8 \mathrm {~ms} ^ { - 1 }\).
  1. Show that \(A B = 7.5 \mathrm {~m}\).
  2. Find the period of the motion.
  3. Find the kinetic energy of \(P\) when it is 2.7 m from \(A\).
  4. Show that the time taken by \(P\) to travel directly from \(A\) to the midpoint of \(O B\) is \(\frac { \pi } { 4 }\) seconds.
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 2003 January Q1
6 marks Standard +0.8
  1. A boy enters a large horizontal field and sees a friend 100 m due north. The friend is walking in an easterly direction at a constant speed of \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The boy can walk at a maximum speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find the shortest time for the boy to intercept his friend and the bearing on which he must travel to achieve this.
(6)
Edexcel M4 2003 January Q2
7 marks Challenging +1.2
2. Boat \(A\) is sailing due cast at a constant speed of \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). To an observer on \(A\), the wind appears to be blowing from due south. A second boat \(B\) is sailing due north at a constant speed of \(14 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). To an observer on \(B\), the wind appears to be blowing from the south west. The velocity of the wind relative to the earth is constant and is the same for both boats. Find the velocity of the wind relative to the earth, stating its magnitude and direction.
Edexcel M4 2003 January Q3
11 marks Challenging +1.8
3. A small pebble of mass \(m\) is placed in a viscous liquid and sinks vertically from rest through the liquid. When the speed of the pebble is \(v\) the magnitude of the resistance due to the liquid is modelled as \(m k v ^ { 2 }\), where \(k\) is a positive constant. Find the speed of the pebble after it has fallen a distance \(D\) through the liquid. \includegraphics[max width=\textwidth, alt={}, center]{618fdb9c-cc0b-4a80-a148-4311c908c94e-3_813_699_397_674} Figure 1 shows a uniform rod \(A B\), of mass \(m\) and length \(4 a\), resting on a smooth fixed sphere of radius \(a\). A light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { 3 } { 4 } m g\), has one end attached to the lowest point \(C\) of the sphere and the other end attached to \(A\). The points \(A\), \(B\) and \(C\) lie in a vertical plane with \(\angle B A C = 2 \theta\), where \(\theta < \frac { \pi } { 4 }\). Given that \(A C\) is always horizontal,
  1. show that the potential energy of the system is $$\frac { m g a } { 8 } \left( 16 \sin 2 \theta + 3 \cot ^ { 2 } \theta - 6 \cot \theta \right) + \text { constant } ,$$
  2. show that there is a value of \(\theta\) for which the system is in equilibrium such that \(0.535 < \theta < 0.545\).
  3. Determine whether this position of equilibrium is stable or unstable.
Edexcel M4 2003 January Q5
17 marks Standard +0.8
5. A particle \(P\) moves in a straight line. At time \(t\) seconds its displacement from a fixed point \(O\) on the line is \(x\) metres. The motion of \(P\) is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 12 \cos 2 t - 6 \sin 2 t$$ When \(t = 0 , P\) is at rest at \(O\).
  1. Find, in terms of \(t\), the displacement of \(P\) from \(O\).
  2. Show that \(P\) comes to instantaneous rest when \(t = \frac { \pi } { 4 }\).
  3. Find, in metres to 3 significant figures, the displacement of \(P\) from \(O\) when \(t = \frac { \pi } { 4 }\).
  4. Find the approximate period of the motion for large values of \(t\). \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{618fdb9c-cc0b-4a80-a148-4311c908c94e-5_534_923_388_541}
    \end{figure} A small ball \(Q\) of mass \(2 m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac { 13 } { 12 } u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d _ { 1 }\) from \(B\), and \(B C\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d _ { 2 }\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(B C\), as shown in Fig. 2, where tan \(\alpha = \frac { 5 } { 12 }\). The line of centres of \(P\) and \(Q\) is parallel to \(B C\). After the collision \(Q\) moves towards \(C\) with speed \(\frac { 3 } { 5 } u\).
  5. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(C B\) are \(\frac { 1 } { 5 } u\) and \(\frac { 5 } { 12 } u\) respectively.
  6. Find the coefficient of restitution between \(P\) and \(Q\).
  7. Show that when \(Q\) reaches \(C , P\) is at a distance \(\frac { 4 } { 3 } d _ { 1 }\) from \(W\). For each collision between a ball and a wall the coefficient of restitution is \(\frac { 1 } { 2 }\).
    Given that the balls collide with each other again,
  8. show that the time between the two collisions of the balls is \(\frac { 15 d _ { 1 } } { u }\),
  9. find the ratio \(d _ { 1 } : d _ { 2 }\). \section*{END}
Edexcel M4 2004 January Q1
5 marks Standard +0.3
  1. A particle \(P\) of mass 3 kg moves in a straight line on a smooth horizontal plane. When the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resultant force acting on \(P\) is a resistance to motion of magnitude \(2 v \mathrm {~N}\). Find the distance moved by \(P\) while slowing down from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (5)
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-2_731_1662_554_227}
\end{figure} Two smooth uniform spheres \(A\) and \(B\) of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of \(A\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When they collide the line joining their centres makes an angle \(\alpha\) with the direction of motion of \(A\) and an angle \(\beta\) with the direction of motion of \(B\), where \(\tan \alpha = \frac { 4 } { 3 }\) and \(\tan \beta = \frac { 12 } { 5 }\) as shown in Fig. 1.
  1. Find the components of the velocities of \(A\) and \(B\) perpendicular and parallel to the line of centres immediately before the collision. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).
  2. Find, to one decimal place, the speed of each sphere after the collision.
    (9)
Edexcel M4 2004 January Q3
14 marks Challenging +1.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-3_933_1063_277_534}
\end{figure} Two uniform rods \(A B\) and \(A C\), each of mass \(2 m\) and length \(2 L\), are freely jointed at \(A\). The mid-points of the rods are \(D\) and \(E\) respectively. A light inextensible string of length \(s\) is fixed to \(E\) and passes round small, smooth light pulleys at \(D\) and \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs vertically. The points \(A , B\) and \(C\) lie in the same vertical plane with \(B\) and \(C\) on a smooth horizontal surface. The angles \(P A B\) and \(P A C\) are each equal to \(\theta ( \theta > 0 )\), as shown in Fig. 2.
  1. Find the length of \(A P\) in terms of \(s , L\) and \(\theta\).
  2. Show that the potential energy \(V\) of the system is given by $$V = 2 m g L ( 3 \cos \theta + \sin \theta ) + \text { constant } .$$
  3. Hence find the value of \(\theta\) for which the system is in equilibrium.
  4. Determine whether this position of equilibrium is stable or unstable.
Edexcel M4 2004 January Q4
14 marks Challenging +1.2
4. A particle \(P\) of mass \(m\) is attached to the mid-point of a light elastic string, of natural length \(2 L\) and modulus of elasticity \(2 m k ^ { 2 } L\), where \(k\) is a positive constant. The ends of the string are attached to points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 3 L\). The particle is released from rest at the point \(C\), where \(A C = 2 L\) and \(A C B\) is a straight line. During the subsequent motion \(P\) experiences air resistance of magnitude \(2 m k v\), where \(v\) is the speed of \(P\). At time \(t , A P = 1.5 L + x\).
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 k ^ { 2 } x = 0\).
  2. Find an expression, in terms of \(t , k\) and \(L\), for the distance \(A P\) at time \(t\).
Edexcel M4 2004 January Q5
14 marks Challenging +1.8
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-4_329_686_999_610}
\end{figure} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point \(X\) with a stationary van of mass 800 kg . After the collision the van came to rest at the point \(A\) having travelled a horizontal distance of 45 m , and the car came to rest at the point \(B\) having travelled a horizontal distance of 21 m . The angle \(A X B\) is \(90 ^ { \circ }\). The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.
  1. Find the coefficient of restitution between the car and the van.
    (5) The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along \(A X\) and the car to a constant horizontal force of 300 N along \(B X\).
  2. Find the speed of the car immediately before the collision.
    (9)
Edexcel M4 2004 January Q6
15 marks Standard +0.8
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-5_431_1090_369_455}
\end{figure} Mary swims in still water at \(0.85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She swims across a straight river which is 60 m wide and flowing at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She sets off from a point \(A\) on the near bank and lands at a point \(B\), which is directly opposite \(A\) on the far bank, as shown in Fig. 4. Find
  1. the angle between the near bank and the direction in which Mary swims,
  2. the time she takes to cross the river. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-5_426_1290_1497_330}
    \end{figure} A little further downstream a large tree has fallen from the far bank into the river. The river is modelled as flowing at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a width of 40 m from the near bank, and \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the 20 m beyond this. Nassim swims at \(0.85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in still water. He swims across the river from a point \(C\) on the near bank. The point \(D\) on the far bank is directly opposite \(C\), as shown in Fig. 5. Nassim swims at the same angle to the near bank as Mary.
  3. Find the maximum distance, downstream from CD, of Nassim during the crossing.
  4. Show that he will land at the point \(D\). \section*{END}
Edexcel M4 2005 January Q2
7 marks Challenging +1.2
2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.] A man cycling at a constant speed \(u\) on horizontal ground 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 constant. When the velocity of the man is \(\frac { u } { 5 } ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), he finds that the velocity of the wind appears to be \(w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(w\) is a constant.
  1. Show that \(v = \frac { u } { 20 }\), and find \(w\) in terms of \(u\).
  2. Find, in terms of \(u\), the true velocity of the wind.
Edexcel M4 2005 January Q3
7 marks Standard +0.8
3. Two ships \(A\) and \(B\) are sailing in the same direction at constant speeds of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and \(16 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) respectively. They are sailing along parallel lines which are 4 km apart. When the distance between the ships is \(4 \mathrm {~km} , B\) turns through \(30 ^ { \circ }\) towards \(A\). Find the shortest distance between the ships in the subsequent motion.
Edexcel M4 2005 January Q4
9 marks Standard +0.3
4. A car of mass \(M\) moves along a straight horizontal road. The total resistance to motion of the car is modelled as having constant magnitude \(R\). The engine of the car works at a constant rate \(R U\). Find the time taken for the car to accelerate from a speed of \(\frac { 1 } { 4 } U\) to a speed of \(\frac { 1 } { 2 } U\). \section*{5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]} The vector \(\mathbf { n } = \left( - \frac { 3 } { 5 } \mathbf { i } + \frac { 4 } { 5 } \mathbf { j } \right)\) and the vector \(\mathbf { p } = \left( - \frac { 4 } { 5 } \mathbf { i } + \frac { 3 } { 5 } \mathbf { j } \right)\) are perpendicular unit vectors.
  1. Verify that \(\frac { 9 } { 5 } \mathbf { n } + \frac { 13 } { 5 } \mathbf { p } = ( \mathbf { i } + 3 \mathbf { j } )\). A smooth uniform sphere \(S\) of mass 0.5 kg is moving on a smooth horizontal plane when it collides with a fixed smooth vertical wall which is parallel to \(\mathbf { p }\). Immediately after the collision the velocity of \(S\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The coefficient of restitution between \(S\) and the wall is \(\frac { 9 } { 16 }\).
  2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(S\) immediately before the collision.
  3. Find the energy lost in the collision. \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d6e5bd56-0a01-44a2-b439-f80cb356d46d-3_681_747_1121_679}
    \end{figure} A smooth wire \(P M Q\) is in the shape of a semicircle with centre \(O\) and radius \(a\). The wire is fixed in a vertical plane with \(P Q\) horizontal and the mid-point \(M\) of the wire vertically below \(O\). A smooth bead \(B\) of mass \(m\) is threaded on the wire and is attached to one end of a light elastic string. The string has modulus of elasticity \(4 m g\) and natural length \(\frac { 5 } { 4 } a\). The other end of the string is attached to a fixed point \(F\) which is a distance \(a\) vertically above \(O\), as shown in Fig. 1.
Edexcel M4 2005 January Q7
18 marks Challenging +1.2
7. A particle of mass \(m\) is attached to one end \(P\) of a light elastic spring \(P Q\), of natural length \(a\) and modulus of elasticity \(m a n ^ { 2 }\). At time \(t = 0\), the particle and the spring are at rest on a smooth horizontal table, with the spring straight but unstretched and uncompressed. The end \(Q\) of the spring is then moved in a straight line, in the direction \(P Q\), with constant acceleration \(f\). At time \(t\), the displacement of the particle in the direction \(P Q\) from its initial position is \(x\) and the length of the spring is \(( a + y )\).
  1. Show that \(x + y = \frac { 1 } { 2 } f t ^ { 2 }\).
  2. Hence show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = \frac { 1 } { 2 } n ^ { 2 } f t ^ { 2 }$$ You are given that the general solution of this differential equation is $$x = A \cos n t + B \sin n t + \frac { 1 } { 2 } f t ^ { 2 } - \frac { f } { n ^ { 2 } }$$ where \(A\) and \(B\) are constants.
  3. Find the values of \(A\) and \(B\).
  4. Find the maximum tension in the spring. END
Edexcel M4 2006 January Q1
7 marks Challenging +1.2
  1. A particle \(P\) of mass 0.5 kg is released from rest at time \(t = 0\) and falls vertically through a liquid. The motion of \(P\) is resisted by a force of magnitude \(2 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(v\) at time \(t\) seconds.
    1. Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 49 - 20 v\).
    2. Find the speed of \(P\) when \(t = 1\).
      (5)
    3. A small smooth sphere \(S\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is attached to a fixed point \(A\) which is at a distance \(a \sqrt { } 3\) from a smooth vertical wall. The sphere \(S\) hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed \(\sqrt { \left( \frac { 37 g a } { 5 } \right) }\).
    4. Show that \(S\) strikes the wall with speed \(\sqrt { \left( \frac { 27 g a } { 5 } \right) }\).
      (4)
    Given that the loss in kinetic energy due to the impact with the wall is \(\frac { 3 m g a } { 5 }\),
  2. find the coefficient of restitution between \(S\) and the wall.
    (7)