Questions - Edexcel M2 - A-Level Maths

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Edexcel M2 2022 January Q5

12 marks Standard +0.3

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.

Explain why $A O = 13 a$ The normal reaction on the rod at $C$ has magnitude $k W$
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.
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.
  1. Explain why $A O = 13 a$

Edexcel M2 2022 January Q6

11 marks Standard +0.3

\hspace{0pt} [The centre of mass of a semicircular arc of radius $r$ is $\frac { 2 r } { \pi }$ from the centre.]

\begin{figure}[h]

[diagram]

\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 $d$.

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.
Find the value of $\alpha$.

Edexcel M2 2022 January Q7

16 marks Standard +0.3

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,
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 $C$
find the speed of the ball at the instant it passes over the pole,
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

8 marks Standard +0.3

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 $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

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 $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
Find the value of $V$

Edexcel M2 2023 January Q2

7 marks Standard +0.3

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

8 marks Standard +0.3

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 $A B D E$.

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.
Find the value of $\theta$, giving your answer to the nearest whole number.

Edexcel M2 2023 January Q4

10 marks Standard +0.3

\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 $\mathbf { i } + \mathbf { j }$

show that $k = 4$
find the magnitude of $\mathbf { F }$ when $t = 2$ Given that $\mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j }$ when $t = 0$
find $\mathbf { r }$ when $t = 2$

Edexcel M2 2023 January Q5

10 marks Standard +0.3

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.

Find the work done against friction as the package moves from $A$ to $B$.
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.
Use the work-energy principle to find the speed of the package at the instant it returns to $A$.

Edexcel M2 2023 January Q6

10 marks Standard +0.3

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 $60 \sqrt { 3 } \mathrm {~N}$,

show that $W = 15$
find the magnitude of the resultant force acting on the pole at $A$.

Edexcel M2 2023 January Q7

10 marks Standard +0.3

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.

Show that $v = \frac { ( 3 - 3 k ) } { k } u$
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 $v \neq u$
find the range of possible values of $k$.

Edexcel M2 2023 January Q8

12 marks Standard +0.3

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.
Use the principle of conservation of mechanical energy to find the value of $U$.
Find the value of $\theta$

Edexcel M2 2024 January Q1

8 marks Moderate -0.8

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

the values of $t$ for which $P$ is instantaneously at rest
the distance travelled by $P$ in the interval $1 \leqslant t \leqslant 3$
the magnitude of the acceleration of $P$ at the instant when $t = 3$

Edexcel M2 2024 January Q2

6 marks Moderate -0.8

\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 $( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }$.

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 $\theta ^ { \circ }$
Find the value of $\theta$

Edexcel M2 2024 January Q3

9 marks Standard +0.3

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.

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.
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

9 marks Challenging +1.2

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 $d$.

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.
Find $F$ in terms of $W$

Edexcel M2 2024 January Q5

9 marks Standard +0.3

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 .

Find the deceleration of the van at this instant.
Find the tension in the towbar at this instant.

Edexcel M2 2024 January Q6

9 marks Standard +0.3

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,

show that $P = 22.4$
find the magnitude of the resultant force acting on the beam at $A$.

Edexcel M2 2024 January Q7

14 marks Standard +0.8

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 $e$.

Find the complete range of possible values of $e$. Given that $e = \frac { 3 } { 5 }$
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 $Q$,
find the complete range of possible values of $f$.

Edexcel M2 2024 January Q8

11 marks Standard +0.3

\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 .

Find the distance $A B$. The speed of $P$ is less than $5 \mathrm {~ms} ^ { - 1 }$ for an interval of length $T$ seconds.
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.
Find the value of $h$.

Edexcel M2 2014 June Q1

11 marks Moderate -0.3

A particle $P$ moves on the $x$-axis. The acceleration of $P$, in the positive $x$ direction at time $t$ seconds, is $( 2 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }$. The velocity of $P$, in the positive $x$ direction at time $t$ seconds, is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $t = 0 , v = 2$
1. Find $v$ in terms of $t$.
The particle is instantaneously at rest at times $t _ { 1 }$ seconds and $t _ { 2 }$ seconds, where $t _ { 1 } < t _ { 2 }$.
Find the values $t _ { 1 }$ and $t _ { 2 }$.
Find the distance travelled by $P$ between $t = t _ { 1 }$ and $t = t _ { 2 }$.

Edexcel M2 2014 June Q2

10 marks Standard +0.3

2. A trailer of mass 250 kg is towed by a car of mass 1000 kg . The car and the trailer are travelling down a straight road inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 20 }$ The resistance to motion of the car is modelled as a single force of magnitude 300 N acting parallel to the road. The resistance to motion of the trailer is modelled as a single force of magnitude 100 N acting parallel to the road. The towbar joining the car to the trailer is modelled as a light rod which is parallel to the direction of motion. At a given instant the car and the trailer are moving down the road with speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and acceleration $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Find the power being developed by the car's engine at this instant.
Find the tension in the towbar at this instant.

Edexcel M2 2014 June Q3

9 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d5f79f83-ccfb-47a5-8100-88db81fd0862-05_1102_732_118_651} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform $\operatorname { rod } A B$ of weight $W$ is freely hinged at end $A$ to a vertical wall. The rod is supported in equilibrium at an angle of $60 ^ { \circ }$ to the wall by a light rigid strut $C D$. The strut is freely hinged to the rod at the point $D$ and to the wall at the point $C$, which is vertically below $A$, as shown in Figure 1. The rod and the strut lie in the same vertical plane, which is perpendicular to the wall. The length of the rod is $4 a$ and $A C = A D = 2.5 a$.

Show that the magnitude of the thrust in the strut is $\frac { 4 \sqrt { 3 } } { 5 } W$.
Find the magnitude of the force acting on the $\operatorname { rod }$ at $A$.

Edexcel M2 2014 June Q4

10 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d5f79f83-ccfb-47a5-8100-88db81fd0862-07_728_748_214_639} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform square lamina $A B C D$ shown in Figure 2 has sides of length 4a. The points $E$ and $F$, on $D A$ and $D C$ respectively, are both at a distance $3 a$ from $D$. The portion $D E F$ of the lamina is folded through $180 ^ { \circ }$ about $E F$ to form the folded lamina $A B C F E$ shown in Figure 3 below. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d5f79f83-ccfb-47a5-8100-88db81fd0862-07_709_730_1395_639} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure}

Show that the distance from $A B$ of the centre of mass of the folded lamina is $\frac { 55 } { 32 } a$.
(6) The folded lamina is freely suspended from $E$ and hangs in equilibrium.
Find the size of the angle between $E D$ and the downward vertical.

Edexcel M2 2014 June Q5

7 marks Moderate -0.3

5. A particle of mass 0.5 kg is moving on a smooth horizontal surface with velocity $12 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it receives an impulse $K ( \mathbf { i } + \mathbf { j } ) \mathrm { N } \mathrm { s }$, where $K$ is a positive constant. Immediately after receiving the impulse the particle is moving with speed $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction which makes an acute angle $\theta$ with the vector $\mathbf { i }$. Find

the value of $K$,
the size of angle $\theta$.

Edexcel M2 2014 June Q6

14 marks Standard +0.3

6. Three particles $P , Q$ and $R$ have masses $3 m , k m$ and 7.5m respectively. The three particles lie at rest in a straight line on a smooth horizontal table with $Q$ between $P$ and $R$. Particle $P$ is projected towards $Q$ with speed $u$ and collides directly with $Q$. The coefficient of restitution between $P$ and $Q$ is $\frac { 1 } { 9 }$.

Show that the speed of $Q$ immediately after the collision is $\frac { 10 u } { 3 ( 3 + k ) }$.
Find the range of values of $k$ for which the direction of motion of $P$ is reversed as a result of the collision. Following the collision between $P$ and $Q$ there is a collision between $Q$ and $R$. Given that $k = 7$ and that $Q$ is brought to rest by the collision with $R$,
find the total kinetic energy lost in the collision between $Q$ and $R$.

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