Questions — Edexcel M2 (551 questions)

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Edexcel M2 2021 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3eb71ecb-fa88-4cca-a2b6-bcf11f1d689b-20_517_947_212_500} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The fixed point \(A\) is 20 m vertically above the point \(O\) which is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The particle moves freely under gravity. At time \(t = 5\) seconds, \(P\) strikes the ground at the point \(B\), where \(O B = 40 \mathrm {~m}\), as shown in Figure 4.
  1. By considering energy, find the speed of \(P\) as it hits the ground at \(B\).
  2. Find the least speed of \(P\) as it moves from \(A\) to \(B\).
  3. Find the length of time for which the speed of \(P\) is more than \(10 \mathrm {~ms} ^ { - 1 }\).
Edexcel M2 2021 January Q8
8. Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately after the collision, \(A\) and \(B\) are moving in the same direction with speeds \(\frac { u } { 3 }\) and \(u\) respectively. In the collision, \(A\) receives an impulse of magnitude 8mu.
  1. Find the coefficient of restitution between \(A\) and \(B\). When \(A\) and \(B\) collide they are at a distance \(d\) from a smooth vertical wall, which is perpendicular to their direction of motion. After the collision with \(A\), particle \(B\) collides directly with the wall and rebounds so that there is a second collision between \(A\) and \(B\). This second collision takes place at distance \(x\) from the wall. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 4 }\)
  2. find \(x\) in terms of \(d\).
    END
Edexcel M2 2022 January Q1
  1. A particle of mass 0.5 kg is moving with velocity \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of ( \(- 4 \mathbf { i } + 6 \mathbf { j }\) )Ns.
    1. Find the speed of the particle immediately after it receives the impulse.
    2. Find the size of the angle between the direction of motion of the particle immediately before it receives the impulse and the direction of motion of the particle immediately after it receives the impulse.
      (3)
Edexcel M2 2022 January Q2
2. A car of mass 600 kg tows a trailer of mass 200 kg up a hill along a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The trailer is attached to the car by a light inextensible towbar. The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude 150 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 300 N . When the engine of the car is working at a constant rate of \(P \mathrm {~kW}\) the car and the trailer have a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the value of \(P\). Later, at the instant when the car and the trailer are travelling up the hill with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. When the towbar breaks the trailer is at the point \(X\). The trailer continues to travel up the hill before coming to instantaneous rest at the point \(Y\). The resistance to the motion of the trailer from non-gravitational forces is again modelled as a constant force of magnitude 300 N .
  2. Use the work-energy principle to find the distance \(X Y\).
    VIIV SIHI NI III M I0N 00 :
Edexcel M2 2022 January Q3
3. A particle \(P\) of mass 0.25 kg is moving on a smooth horizontal surface under the action of a single force, \(\mathbf { F }\) newtons. At time \(t\) seconds \(( t \geqslant 0 )\), the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = ( 6 \sin 3 t ) \mathbf { i } + ( 1 + 2 \cos t ) \mathbf { j }$$
  1. Find \(\mathbf { F }\) in terms of \(t\). At time \(t = 0\), the position vector of \(P\) relative to a fixed point \(O\) is \(( 4 \mathbf { i } - \sqrt { 3 } \mathbf { j } ) \mathrm { m }\).
  2. Find the position vector of \(P\) relative to \(O\) when \(P\) is first moving parallel to the vector \(\mathbf { i }\).
Edexcel M2 2022 January Q4
4. Two small balls, \(A\) and \(B\), are moving in opposite directions along the same straight line on smooth horizontal ground. The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(3 m\). The balls collide directly. Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\) By modelling the balls as particles,
  1. show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 5 } u ( 1 + 6 e )\).
    (6) After the collision with ball \(A\), ball \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 5 } { 7 }\)
    Ball \(B\) rebounds from the wall and there is a second direct collision between \(A\) and \(B\).
  2. Find the range of possible values of \(e\).
Edexcel M2 2022 January Q5
5. A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5 a\).
A uniform rod \(A B\), of length \(16 a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(A C = 12 a\) and angle \(C A O = \alpha\), as shown in Figure 1. Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
  1. Explain why \(A O = 13 a\) The normal reaction on the rod at \(C\) has magnitude \(k W\)
  2. Show that \(k = \frac { 8 } { 13 }\) The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta ^ { \circ }\) to the horizontal.
  3. Find
    1. an expression for \(R\) in terms of \(W\)
    2. the value of \(\theta\)
      (8) 5 \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-16_426_1001_125_475}
      \end{figure} . T a and angle \(C A O = \alpha\), as shown in Figure 1.
      Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
  4. Explain why \(A O = 13 a\)
Edexcel M2 2022 January Q6
  1. \hspace{0pt} [The centre of mass of a semicircular arc of radius \(r\) is \(\frac { 2 r } { \pi }\) from the centre.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-20_668_371_358_790} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework,
  • \(A B C\) is straight and has length \(25 a\)
  • \(A D E\) is straight and has length \(24 a\)
  • \(A B D\) is a semicircular arc of radius \(7 a\)
  • \(E C = 7 a\)
  • angle \(A E C = 90 ^ { \circ }\)
  • the points \(A , B , C , D\) and \(E\) all lie in the same plane
The distance of the centre of mass of the framework from \(A E\) is \(d\).
  1. Show that \(d = \frac { 53 } { 2 ( 7 + \pi ) } a\) The framework is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at angle \(\alpha ^ { \circ }\) to the downward vertical.
  2. Find the value of \(\alpha\).
Edexcel M2 2022 January Q7
  1. A particle \(P\) is projected from a fixed point \(O\) on horizontal ground. The particle is projected with speed \(u\) at an angle \(\alpha\) above the horizontal. At the instant when the horizontal distance of \(P\) from \(O\) is \(x\), the vertical distance of \(P\) above the ground is \(y\). The motion of \(P\) is modelled as that of a particle moving freely under gravity.
    1. Show that \(y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)\)
      (6)
    A small ball is projected from the fixed point \(O\) on horizontal ground. The ball is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta ^ { \circ }\) above the horizontal. A vertical pole \(A B\), of height 2 m , stands on the ground with \(O A = 10 \mathrm {~m}\), as shown in Figure 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-24_246_899_840_525} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The ball is modelled as a particle moving freely under gravity and the pole is modelled as a rod.
    The path of the ball lies in the vertical plane containing \(O , A\) and \(B\).
    Using the model,
  2. find the range of values of \(\theta\) for which the ball will pass over the pole. Given that \(\theta = 40\) and that the ball first hits the ground at the point \(C\)
  3. find the speed of the ball at the instant it passes over the pole,
  4. find the distance \(O C\). \includegraphics[max width=\textwidth, alt={}, center]{0762451f-b951-4d66-9e01-61ecb7b30d95-28_2649_1898_109_169}
Edexcel M2 2023 January Q1
  1. A truck of mass 1500 kg is moving on a straight horizontal road.
The engine of the truck is working at a constant rate of 30 kW .
The resistance to the motion of the truck is modelled as a constant force of magnitude \(R\) newtons.
At the instant when the truck is moving at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Find the value of \(R\). Later on, the truck is moving up a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 8 }\) The resistance to the motion of the truck from non-gravitational forces is modelled as a constant force of magnitude 500 N .
    The engine of the truck is again working at a constant rate of 30 kW . At the instant when the speed of the truck is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the deceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  2. Find the value of \(V\)
Edexcel M2 2023 January Q2
  1. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) The particle receives an impulse \(( - 2 \mathbf { i } + \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after receiving the impulse, the velocity of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The kinetic energy gained by \(P\) as a result of receiving the impulse is 22 J .
Find the possible values of \(\lambda\).
Edexcel M2 2023 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee5f81bc-1bdb-47a1-81e7-7e3cb8219e91-06_618_803_244_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(A B D E\) is in the shape of a rectangle with \(A B = 8 a\) and \(B D = 6 a\). The triangle \(B C D\) is isosceles and has base \(6 a\) and perpendicular height \(6 a\). The template \(A B C D E\), shown shaded in Figure 1, is formed by removing the triangular lamina \(B C D\) from the lamina \(A B D E\).
  1. Show that the centre of mass of the template is \(\frac { 14 } { 5 } a\) from \(A E\). The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.
  2. Find the value of \(\theta\), giving your answer to the nearest whole number.
Edexcel M2 2023 January Q4
  1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.]
A particle \(Q\) of mass 1.5 kg is moving on a smooth horizontal plane under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds ( \(t \geqslant 0\) ), the position vector of \(Q\), relative to a fixed point \(O\), is \(\mathbf { r }\) metres and the velocity of \(Q\) is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\)
It is given that $$\mathbf { v } = \left( 3 t ^ { 2 } + 2 t \right) \mathbf { i } + \left( t ^ { 3 } + k t \right) \mathbf { j }$$ where \(k\) is a constant.
Given that when \(t = 2\) particle \(Q\) is moving in the direction of the vector \(\mathbf { i } + \mathbf { j }\)
  1. show that \(k = 4\)
  2. find the magnitude of \(\mathbf { F }\) when \(t = 2\) Given that \(\mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j }\) when \(t = 0\)
  3. find \(\mathbf { r }\) when \(t = 2\)
Edexcel M2 2023 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee5f81bc-1bdb-47a1-81e7-7e3cb8219e91-12_296_1125_246_470} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) The points \(A\) and \(B\) are on a line of greatest slope of the ramp, with \(A B = 2.5 \mathrm {~m}\) and \(B\) above \(A\), as shown in Figure 2. A package of mass 1.5 kg is projected up the ramp from \(A\) with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and first comes to instantaneous rest at \(B\). The coefficient of friction between the package and the ramp is \(\frac { 2 } { 7 }\)
The package is modelled as a particle.
  1. Find the work done against friction as the package moves from \(A\) to \(B\).
  2. Use the work-energy principle to find the value of \(U\). After coming to instantaneous rest at \(B\), the package slides back down the slope.
  3. Use the work-energy principle to find the speed of the package at the instant it returns to \(A\).
Edexcel M2 2023 January Q6
6. Figure 3 A uniform pole \(A B\), of weight 50 N and length 6 m , has a particle of weight \(W\) newtons attached at its end \(B\). The pole has its end \(A\) freely hinged to a vertical wall.
A light rod holds the particle and pole in equilibrium with the pole at \(60 ^ { \circ }\) to the wall. One end of the light rod is attached to the pole at \(C\), where \(A C = 4 \mathrm {~m}\).
The other end of the light rod is attached to the wall at the point \(D\).
The point \(D\) is vertically below \(A\) with \(A D = 4 \mathrm {~m}\), as shown in Figure 3.
The pole and the light rod lie in a vertical plane which is perpendicular to the wall.
The pole is modelled as a rod.
Given that the thrust in the light rod is \(60 \sqrt { 3 } \mathrm {~N}\),
  1. show that \(W = 15\)
  2. find the magnitude of the resultant force acting on the pole at \(A\).
Edexcel M2 2023 January Q7
  1. Particle \(P\) has mass \(3 m\) and particle \(Q\) has mass \(k m\). The particles are moving towards each other on the same straight line on a smooth horizontal surface.
    The particles collide directly.
    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 \(u\) and the speed of \(Q\) is \(v\).
The direction of motion of \(P\) is unchanged by the collision.
  1. Show that \(v = \frac { ( 3 - 3 k ) } { k } u\)
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    Given that \(v \neq u\)
  3. find the range of possible values of \(k\).
Edexcel M2 2023 January Q8
  1. A particle \(P\) is projected from a fixed point \(O\). The particle is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\alpha\) above the horizontal. The particle moves freely under gravity. At the instant when the horizontal distance of \(P\) from \(O\) is \(x\) metres, \(P\) is \(y\) metres vertically above the level of \(O\).
    1. Show that \(y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)\)
    A small ball is projected from a fixed point \(A\) with speed \(U \mathrm {~ms} ^ { - 1 }\) at \(\theta ^ { \circ }\) above the horizontal.
    The point \(B\) is on horizontal ground and is vertically below the point \(A\), with \(A B = 20 \mathrm {~m}\).
    The ball hits the ground at the point \(C\), where \(B C = 30 \mathrm {~m}\), as shown in Figure 4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ee5f81bc-1bdb-47a1-81e7-7e3cb8219e91-24_556_961_904_552} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The speed of the ball immediately before it hits the ground is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    The motion of the ball is modelled as that of a particle moving freely under gravity.
  2. Use the principle of conservation of mechanical energy to find the value of \(U\).
  3. Find the value of \(\theta\)
Edexcel M2 2024 January Q1
  1. A particle \(P\) moves along a straight line. The fixed point \(O\) is on the line. At time \(t\) seconds, \(t > 0\), the displacement of \(P\) from \(O\) is \(x\) metres, where
$$x = 2 t ^ { 3 } - 21 t ^ { 2 } + 60 t$$ Find
  1. the values of \(t\) for which \(P\) is instantaneously at rest
  2. the distance travelled by \(P\) in the interval \(1 \leqslant t \leqslant 3\)
  3. the magnitude of the acceleration of \(P\) at the instant when \(t = 3\)
Edexcel M2 2024 January Q2
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]
A particle \(Q\) of mass 0.5 kg is moving on a smooth horizontal surface. Particle \(Q\) is moving with velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }\).
  1. Find the speed of \(Q\) immediately after receiving the impulse. As a result of receiving the impulse, the direction of motion of \(Q\) is turned through an angle \(\theta ^ { \circ }\)
  2. Find the value of \(\theta\)
Edexcel M2 2024 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f06fe7-4d9c-4009-8931-3ecbc31fa5e5-06_323_1043_255_513} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 7 }\)
The line \(A B\) is a line of greatest slope of the ramp, with \(B\) above \(A\) and \(A B = 6 \mathrm {~m}\), as shown in Figure 1. A block \(P\) of mass 2 kg is pushed, with constant speed, in a straight line up the slope from \(A\) to \(B\). The force pushing \(P\) acts parallel to \(A B\). The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 3 }\)
The block is modelled as a particle and air resistance is negligible.
  1. Use the model to find the total work done in pushing the block from \(A\) to \(B\). The block is now held at \(B\) and released from rest.
  2. Use the model and the work-energy principle to find the speed of the block at the instant it reaches \(A\).
Edexcel M2 2024 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f06fe7-4d9c-4009-8931-3ecbc31fa5e5-10_552_680_255_447} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f06fe7-4d9c-4009-8931-3ecbc31fa5e5-10_547_494_255_1165} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The uniform rectangular lamina \(A B C D\), shown in Figure 2, has \(D C = 4 a\) and \(A D = 5 a\)
The points \(S\) on \(A B\) and \(T\) on \(B C\) are such that \(S B = B T = 3 a\)
The lamina is folded along \(S T\) to form the folded lamina \(L\), shown in Figure 3.
The distance of the centre of mass of \(L\) from \(A D\) is \(d\).
  1. Show that \(d = \frac { 71 } { 40 } a\) The weight of \(L\) is \(4 W\). A particle of weight \(W\) is attached to \(L\) at \(C\).
    The folded lamina \(L\) is freely suspended from \(S\).
    A force of magnitude \(F\), acting parallel to \(D C\), is applied to \(L\) at \(D\) so that \(A D\) is vertical.
  2. Find \(F\) in terms of \(W\)
Edexcel M2 2024 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f06fe7-4d9c-4009-8931-3ecbc31fa5e5-14_355_1230_244_422} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A van of mass 600 kg is moving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\). The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 4. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 250 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 150 N . The towbar is modelled as a light rod.
At the instant when the speed of the van is \(16 \mathrm {~ms} ^ { - 1 }\), the engine of the van is working at a rate of 10 kW .
  1. Find the deceleration of the van at this instant.
  2. Find the tension in the towbar at this instant.
Edexcel M2 2024 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f06fe7-4d9c-4009-8931-3ecbc31fa5e5-18_424_990_255_539} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A uniform beam \(A B\), of weight 40 N and length 7 m , rests with end \(A\) on rough horizontal ground. The beam rests on a smooth horizontal peg at \(C\), with \(A C = 5 \mathrm {~m}\), as shown in Figure 5.
The beam is inclined at an angle \(\theta\) to the ground, where \(\sin \theta = \frac { 3 } { 5 }\)
The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg.
The normal reaction between the beam and the peg at \(C\) has magnitude \(P\) newtons.
Using the model,
  1. show that \(P = 22.4\)
  2. find the magnitude of the resultant force acting on the beam at \(A\).
Edexcel M2 2024 January Q7
  1. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(5 m\).
The particles are moving in the same direction 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 \(6 u\) and the speed of \(Q\) is \(u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(P\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Find the complete range of possible values of \(e\). Given that \(e = \frac { 3 } { 5 }\)
  2. find the total kinetic energy lost in the collision between \(P\) and \(Q\). 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 a second collision between \(P\) and \(Q\),
  3. find the complete range of possible values of \(f\).
Edexcel M2 2024 January Q8
  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]{d5f06fe7-4d9c-4009-8931-3ecbc31fa5e5-26_273_889_296_589} \captionsetup{labelformat=empty} \caption{Figure 6}
\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 \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)
Particle \(P\) moves freely under gravity and hits the ground at \(B\), as shown in Figure 6 .
  1. Find the distance \(A B\). The speed of \(P\) is less than \(5 \mathrm {~ms} ^ { - 1 }\) for an interval of length \(T\) seconds.
  2. Find the value of \(T\) At the instant when the direction of motion of \(P\) is perpendicular to the initial direction of motion of \(P\), the particle is \(h\) metres above the ground.
  3. Find the value of \(h\).