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Edexcel M2 2018 October Q6
10 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-20_755_579_267_703} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A uniform rod, \(A B\), of mass \(8 m\) and length \(2 a\), has its end \(A\) resting against a rough vertical wall. One end of a light inextensible string is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\), which is vertically above \(A\). The angle between the rod and the string is \(30 ^ { \circ }\). A particle of mass \(k m\) is fixed to the rod at \(C\), where \(A C = 0.5 a\). The rod is in equilibrium in a vertical plane perpendicular to the wall, and is at an angle of \(60 ^ { \circ }\) to the wall, as shown in Figure 5. The tension in the string is \(T\).
  1. Show that \(T = \frac { \sqrt { 3 } } { 4 } ( 16 + k ) m g\) The coefficient of friction between the wall and the rod is \(\frac { 2 } { 3 } \sqrt { 3 }\).
    Given that the rod is in limiting equilibrium,
  2. find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-23_67_65_2656_1886}
Edexcel M2 2018 October Q7
16 marks Standard +0.8
7. A particle, \(P\), of mass \(k m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal surface. Particle \(P\) collides directly with another particle, \(Q\), of mass \(2 m\) which is moving with speed \(u\) in the same direction along the same straight line. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Given that immediately after the collision \(P\) and \(Q\) are moving in opposite directions and the speed of \(Q\) is \(\frac { 3 } { 2 } u\),
  1. find the range of possible values of \(e\). It is now also given that \(e = \frac { 7 } { 8 }\).
  2. Show that the kinetic energy lost by \(P\) in the collision with \(Q\) is \(\frac { 11 } { 8 } m u ^ { 2 }\). The collision between \(P\) and \(Q\) takes place at the point \(A\). After the collision, \(Q\) hits a fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The distance from \(A\) to the wall is \(d\). The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\). Particle \(Q\) rebounds from the wall and moves so that \(P\) and \(Q\) collide directly at the point \(B\).
  3. Find, in terms of \(d\) and \(u\), the time interval between the collision at \(A\) and the collision at \(B\).
    \includegraphics[max width=\textwidth, alt={}]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-28_2639_1833_121_118}
Edexcel M2 2021 October Q1
7 marks Standard +0.3
1. \section*{Figure 1} Figure 1 A particle of mass \(m\) is held at rest at a point \(A\) on a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) The coefficient of friction between the particle and the plane is \(\frac { 1 } { 5 }\) The points \(A\) and \(B\) lie on a line of greatest slope of the plane, with \(B\) above \(A\), and \(A B = d\), as shown in Figure 1. The particle is pushed up the line of greatest slope from \(A\) to \(B\).
  1. Show that the work done against friction as the particle moves from \(A\) to \(B\) is \(\frac { 12 } { 65 } m g d\) The particle is then held at rest at \(B\) and released.
  2. Use the work-energy principle to find, in terms of \(g\) and \(d\), the speed of the particle at the instant it reaches \(A\).
Edexcel M2 2021 October Q2
8 marks Standard +0.3
2. A vehicle of mass 450 kg is moving on a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\) At the instant when the vehicle is moving down the road at \(12 \mathrm {~ms} ^ { - 1 }\)
  • the engine of the vehicle is working at a rate of \(P\) watts
  • the acceleration of the vehicle is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  • the resistance to the motion of the vehicle is modelled as a constant force of magnitude \(R\) newtons
At the instant when the vehicle is moving up the road at \(12 \mathrm {~ms} ^ { - 1 }\)
  • the engine of the vehicle is working at a rate of \(2 P\) watts
  • the deceleration of the vehicle is \(0.5 \mathrm {~ms} ^ { - 2 }\)
  • the resistance to the motion of the vehicle from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons
Find the value of \(P\).
Edexcel M2 2021 October Q3
9 marks Standard +0.3
3. A particle \(P\) moves on the \(x\)-axis. At time \(t = 0 , P\) is instantaneously at rest at \(O\).
At time \(t\) seconds, \(t > 0\), the \(x\) coordinate of \(P\) is given by $$x = 2 t ^ { \frac { 7 } { 2 } } - 14 t ^ { \frac { 5 } { 2 } } + \frac { 56 } { 3 } t ^ { \frac { 3 } { 2 } }$$ Find
  1. the non-zero values of \(t\) for which \(P\) is at instantaneous rest
  2. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)
  3. the acceleration of \(P\) when \(t = 4\) \(\_\_\_\_\)}
Edexcel M2 2021 October Q4
6 marks Standard +0.3
4. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(\mathbf { J }\) Ns. Immediately after \(P\) receives the impulse, the speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Given that \(\mathbf { J } = c ( - \mathbf { i } + 2 \mathbf { j } )\), where \(c\) is a constant, find the two possible values of \(c\).
(6)
Edexcel M2 2021 October Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-12_597_502_210_721} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A pole \(A B\) has length 2.5 m and weight 70 N .
The pole rests with end \(B\) against a rough vertical wall. One end of a cable of length 4 m is attached to the pole at \(A\). The other end of the cable is attached to the wall at the point \(C\). The point \(C\) is vertically above \(B\) and \(B C = 2.5 \mathrm {~m}\).
The angle between the cable and the wall is \(\alpha\), as shown in Figure 2.
The pole is in a vertical plane perpendicular to the wall.
The cable is modelled as a light inextensible string and the pole is modelled as a uniform rod. Given that \(\tan \alpha = \frac { 3 } { 4 }\)
  1. show that the tension in the cable is 56 N . Given also that the pole is in limiting equilibrium,
  2. find the coefficient of friction between the pole and the wall. \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-15_90_61_2613_1886}
Edexcel M2 2021 October Q6
10 marks Standard +0.8
6. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(3 m\).
Immediately after the collision, \(A\) and \(B\) are moving in opposite directions with the same speed \(v\).
In the collision, \(A\) receives an impulse of magnitude \(5 m v\).
  1. Find the coefficient of restitution between \(A\) and \(B\).
    (6) After the collision with \(A\), particle \(B\) strikes a smooth fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of the particles.
    The coefficient of restitution between \(B\) and the wall is \(f\).
    As a result of its collision with \(A\) and with the wall, the total kinetic energy lost by \(B\) is \(E\). As a result of its collision with \(B\), the kinetic energy lost by \(A\) is \(2 E\).
  2. Find the value of \(f\). \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-19_2664_107_106_6}
    "
    , \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-19_108_67_2613_1884}
Edexcel M2 2021 October Q7
11 marks Standard +0.3
7. In this question you may use, without proof, the formula for the centre of mass of a uniform sector of a circle, as given in the formulae book. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-20_444_625_354_662} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The uniform lamina \(A B C D E\), shown shaded in Figure 3, is formed by joining a rectangle to a sector of a circle.
  • The rectangle \(A B C E\) has \(A B = E C = a\) and \(A E = B C = d\)
  • The sector \(C D E\) has centre \(C\) and radius \(a\)
  • Angle \(E C D = \frac { \pi } { 3 }\) radians
The centre of mass of the lamina lies on EC.
  1. Show that \(a = \sqrt { 3 } d\) The lamina is freely suspended from \(B\) and hangs in equilibrium with \(B C\) at an angle \(\beta\) radians to the downward vertical.
  2. Find the value of \(\beta\)
Edexcel M2 2021 October Q8
14 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-24_470_824_214_561} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The fixed point \(A\) is \(h\) metres vertically above the point \(O\) that is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle moves freely under gravity. At time \(t = 2.5\) seconds, \(P\) strikes the ground at the point \(B\). At the instant when \(P\) strikes the ground, the speed of \(P\) is \(18 \mathrm {~ms} ^ { - 1 }\), as shown in Figure 4.
  1. By considering energy, find the value of \(h\).
  2. Find the distance \(O B\). As \(P\) moves from \(A\) to \(B\), the speed of \(P\) is less than or equal to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds.
  3. Find the value of \(T\)
Edexcel M2 2022 October Q1
6 marks Moderate -0.8
  1. Three particles of masses \(2 m , 3 m\) and \(4 m\) are placed at the points with coordinates \(( - 2,5 ) , ( 2 , - 3 )\) and \(( 3 k , k )\) respectively, where \(k\) is a constant. The centre of mass of the three particles is at the point \(( \bar { x } , \bar { y } )\).
    1. Show that \(\bar { x } = \frac { 2 + 12 k } { 9 }\)
    The centre of mass of the three particles lies at a point on the straight line with equation \(x + 2 y = 3\)
  2. Find the value of \(k\).
Edexcel M2 2022 October Q2
5 marks Standard +0.3
2. A car of mass 900 kg is moving down a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\) The engine of the car is working at a constant rate of 15 kW .
The resistance to the motion of the car is modelled as a constant force of magnitude 400 N . Find the acceleration of the car at the instant when it is moving at \(16 \mathrm {~ms} ^ { - 1 }\)
VIAV SIHI NI IIIIIM ION OCVIIIV SIHI NI III IM I O N OCVIIV SIHI NI IIIIM I I ON OC
\section*{Qu}
Edexcel M2 2022 October Q3
6 marks Standard +0.3
  1. A particle \(P\) of mass 0.2 kg is moving with velocity \(( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)
The particle receives an impulse \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { Ns }\), where \(\lambda\) is a constant.
Immediately after receiving the impulse, the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the possible values of \(\lambda\)
Edexcel M2 2022 October Q4
10 marks Standard +0.3
4. At time \(t\) seconds \(( 0 \leqslant t < 5 )\), a particle \(P\) has velocity \(\mathbf { v m s } ^ { - 1 }\), where $$\mathbf { v } = ( \sqrt { 5 - t } ) \mathbf { i } + \left( t ^ { 2 } + 2 t - 3 \right) \mathbf { j }$$ When \(t = \lambda\), particle \(P\) is moving in a direction parallel to the vector \(\mathbf { i }\).
  1. Find the acceleration of \(P\) when \(t = \lambda\) The position vector of \(P\) is measured relative to the fixed point \(O\) When \(t = 1\), the position vector of \(P\) is \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { m }\). Given that \(1 \leqslant T < 5\)
  2. find, in terms of \(T\), the position vector of \(P\) when \(t = T\)
Edexcel M2 2022 October Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-12_424_1118_221_420} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\) has length \(8 a\) and weight \(W\).
The end \(A\) of the rod is freely hinged to horizontal ground.
The rod rests in equilibrium against a block which is also fixed to the ground.
The block is modelled as a smooth solid hemisphere with radius \(2 a\) and centre \(D\).
The point of contact between the rod and the block is \(C\), where \(A C = 5 a\) The rod is at an angle \(\theta\) to the ground, as shown in Figure 1.
Points \(A , B , C\) and \(D\) all lie in the same vertical plane.
  1. Show that \(A D = \sqrt { 29 } a\)
  2. Show that the magnitude of the normal reaction at \(C\) between the rod and the block is \(\frac { 4 } { \sqrt { 29 } } W\) The resultant force acting on the rod at \(A\) has magnitude \(k W\) and acts at an angle \(\alpha\) to the ground.
  3. Find (i) the exact value of \(k\) (ii) the exact value of \(\tan \alpha\)
    \includegraphics[max width=\textwidth, alt={}, center]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-15_72_1819_2709_114}
Edexcel M2 2022 October Q6
9 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-16_588_871_219_539} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(P Q R S T U V\) shown in Figure 2 is formed from two identical rectangles, \(P Q U V\) and \(Q R S T U\).
The rectangles have sides \(P Q = R S = 2 a\) and \(P V = Q R = k a\).
  1. Show that the centre of mass of the lamina is \(\left( \frac { 6 + k } { 4 } \right) a\) from \(P V\) The lamina is freely suspended from \(P\) and hangs in equilibrium with \(P R\) at an angle of \(\alpha\) to the downward vertical. Given that \(\tan \alpha = \frac { 7 } { 15 }\)
  2. find the value of \(k\).
Edexcel M2 2022 October Q7
13 marks Standard +0.3
7. Particle \(A\) has mass \(m\) and particle \(B\) has mass \(2 m\). The particles are moving in the same direction along the same straight line on a smooth horizontal surface.
Particle \(A\) collides directly with particle \(B\).
Immediately before the collision, the speed of \(A\) is \(3 u\) and the speed of \(B\) is \(u\).
The coefficient of restitution between \(A\) and \(B\) is \(e\).
    1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 + 2 e } { 3 } u\)
    2. Find the speed of \(A\) immediately after the collision. After the collision, \(B\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(B\).
      The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 3 }\) Particle \(B\) rebounds and there is a second collision between \(A\) and \(B\).
      The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall.
      The time between the two collisions is \(T\).
      Given that \(e = \frac { 1 } { 2 }\)
  2. find \(T\) in terms of \(d\) and \(u\).
Edexcel M2 2022 October Q8
14 marks Standard +0.3
8. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-24_378_1219_347_349} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\) The point \(A\) is at the bottom of the ramp and the point \(B\) is at the top of the ramp. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 15 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 0.3 kg is projected with speed \(U \mathrm {~ms} ^ { - 1 }\) from \(A\) directly towards \(B\). At the instant \(P\) reaches the point \(B\), the velocity of \(P\) is \(( 24 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the horizontal ground at the point \(C\).
The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 5 }\)
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
  2. Use the work-energy principle to find the value of \(U\).
  3. Find the time taken by \(P\) to move from \(B\) to \(C\). At the instant immediately before \(P\) hits the ground at \(C\), the particle is moving downwards at \(\theta ^ { \circ }\) to the horizontal.
  4. Find the value of \(\theta\)
Edexcel M2 2023 October Q1
7 marks Standard +0.3
  1. At time \(t\) seconds, \(t > 0\), a particle \(P\) is at the point with position vector \(\mathbf { r } \mathrm { m }\), where
$$\mathbf { r } = \left( t ^ { 4 } - 8 t ^ { 2 } \right) \mathbf { i } + \left( 6 t ^ { 2 } - 2 t ^ { \frac { 3 } { 2 } } \right) \mathbf { j }$$
  1. Find the velocity of \(P\) when \(P\) is moving in a direction parallel to the vector \(\mathbf { j }\)
  2. Find the acceleration of \(P\) when \(t = 4\)
Edexcel M2 2023 October Q2
14 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-04_784_814_260_646} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a template where
  • PQUY is a uniform square lamina with sides of length \(4 a\)
  • RSTU is a uniform square lamina with sides of length \(2 a\)
  • VWXY is a uniform square lamina with sides of length \(2 a\)
  • the three squares all lie in the same plane
  • the mass per unit area of \(V W X Y\) is double the mass per unit area of \(P Q U Y\)
  • the mass per unit area of \(R S T U\) is double the mass per unit area of \(P Q U Y\)
  • the distance of the centre of mass of the template from \(P X\) is \(d\)
    1. Show that \(d = \frac { 5 } { 2 } a\)
The template is freely pivoted about \(Q\) and hangs in equilibrium with \(P Q\) at an angle of \(\theta\) to the downward vertical.
  • Find the value of \(\tan \theta\) The mass of the template is \(M\) The template is still freely pivoted about \(Q\), but it is now held in equilibrium, with \(P Q\) vertical, by a horizontal force of magnitude \(F\) which acts on the template at \(X\). The line of action of the force lies in the same plane as the template.
  • Find \(F\) in terms of \(M\) and \(g\)
    •
    Edexcel M2 2023 October Q3
    6 marks Standard +0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-08_424_752_246_667} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A particle \(Q\) of mass 0.25 kg is moving in a straight line on a smooth horizontal surface with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\). The impulse acts parallel to the horizontal surface and at \(60 ^ { \circ }\) to the original direction of motion of \(Q\). Immediately after receiving the impulse, the speed of \(Q\) is \(12 \mathrm {~ms} ^ { - 1 }\) As a result of receiving the impulse, the direction of motion of \(Q\) is turned through \(\alpha ^ { \circ }\), as shown in Figure 2. Find the value of \(I\)
    Edexcel M2 2023 October Q4
    12 marks Standard +0.3
    1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors, with \(\mathbf { i }\) horizontal and \(\mathbf { j }\) vertical.]
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-12_278_891_294_587} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The fixed points \(A\) and \(B\) lie on horizontal ground.
    At time \(t = 0\), a particle \(P\) is projected from \(A\) with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) Particle \(P\) moves freely under gravity and hits the ground at \(B\), as shown in Figure 3 .
    At time \(T _ { 1 }\) seconds, \(P\) is at its highest point above the ground.
    1. Find the value of \(T _ { 1 }\) At time \(t = 0\), a particle \(Q\) is also projected from \(A\) but with velocity \(( 5 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Particle \(Q\) moves freely under gravity.
    2. Find the vertical distance between \(Q\) and \(P\) at time \(T _ { 1 }\) seconds, giving your answer to 2 significant figures. At the instant when particle \(P\) reaches \(B\), particle \(Q\) is moving at \(\alpha ^ { \circ }\) below the horizontal.
    3. Find the value of \(\alpha\). At time \(T _ { 2 }\) seconds, the direction of motion of \(Q\) is perpendicular to the initial direction of motion of \(Q\).
    4. Find the value of \(T _ { 2 }\)
    Edexcel M2 2023 October Q5
    13 marks Standard +0.3
    1. A cyclist is travelling on a straight horizontal road and working at a constant rate of 500 W .
    The total mass of the cyclist and her cycle is 80 kg .
    The total resistance to the motion of the cyclist is modelled as a constant force of magnitude 60 N .
    1. Using this model, find the acceleration of the cyclist at the instant when her speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) On the following day, the cyclist travels up a straight road from a point \(A\) to a point \(B\).
      The distance from \(A\) to \(B\) is 20 km .
      Point \(A\) is 500 m above sea level and point \(B\) is 800 m above sea level.
      The cyclist starts from rest at \(A\).
      At the instant she reaches \(B\) her speed is \(8 \mathrm {~ms} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude 60 N .
    2. Using this model, find the total work done by the cyclist in the journey from \(A\) to \(B\). Later on, the cyclist is travelling up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\) The cyclist is now working at a constant rate of \(P\) watts and has a constant speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is again modelled as a constant force of magnitude 60 N .
    3. Using this model, find the value of \(P\)
    Edexcel M2 2023 October Q6
    9 marks Standard +0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-20_593_745_246_667} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A uniform \(\operatorname { rod } A B\) has length \(8 a\) and weight \(W\).
    The end \(A\) of the rod is freely hinged to a fixed point on a vertical wall.
    A particle of weight \(\frac { 1 } { 4 } W\) is attached to the rod at \(B\).
    A light inelastic string of length \(5 a\) has one end attached to the rod at the point \(C\), where \(A C = 5 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is above \(A\) and \(A D = 5 a\), as shown in Figure 4. The rod rests in equilibrium.
    The tension in the string is \(T\).
    1. Show that \(T = \frac { 6 } { 5 } \mathrm {~W}\)
    2. Find, in terms of \(W\), the magnitude of the force exerted on the rod by the hinge at \(A\).
    Edexcel M2 2023 October Q7
    14 marks Standard +0.3
    1. Particle \(P\) has mass \(4 m\) and particle \(Q\) has mass \(2 m\).
    The particles are moving in opposite directions along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
    Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\).
    Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
    The direction of motion of each particle is reversed as a result of the collision.
    The total kinetic energy of \(P\) and \(Q\) after the collision is half of the total kinetic energy of \(P\) and \(Q\) before the collision.
    1. Show that \(y = \frac { 8 } { 3 } u\) The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    2. Find the value of \(e\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
      The coefficient of restitution between \(Q\) and the wall is \(f\).
      Given that there is no second collision between \(P\) and \(Q\),
    3. find the range of possible values of \(f\). Given that \(f = \frac { 1 } { 4 }\)
    4. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) as a result of its impact with the wall.