Questions - Edexcel - A-Level Maths

A smooth uniform sphere $P$ has mass 0.3 kg . Another smooth uniform sphere $Q$, with the same radius as $P$, has mass 0.2 kg .

The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of $P$ is $( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. At the instant of collision, the line joining the centres of the spheres is parallel to $\mathbf { i }$.
The kinetic energy of $Q$ immediately after the collision is half the kinetic energy of $Q$ immediately before the collision.

Find
1. the velocity of $P$ immediately after the collision,
2. the velocity of $Q$ immediately after the collision,
3. the coefficient of restitution between $P$ and $Q$,
  carefully justifying your answers.
Find the size of the angle through which the direction of motion of $P$ is deflected by the collision.

Edexcel FM1 2020 June Q6

11 marks Challenging +1.2

A light elastic string with natural length $l$ and modulus of elasticity $k m g$ has one end attached to a fixed point $A$ on a rough inclined plane. The other end of the string is attached to a package of mass $m$.

The plane is inclined at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 5 } { 12 }$
The package is initially held at $A$. The package is then projected with speed $\sqrt { 6 g l }$ up a line of greatest slope of the plane and first comes to rest at the point $B$, where $A B = 31$.
The coefficient of friction between the package and the plane is $\frac { 1 } { 4 }$
By modelling the package as a particle,

show that $k = \frac { 15 } { 26 }$
find the acceleration of the package at the instant it starts to move back down the plane from the point $B$.

Edexcel FM1 2020 June Q7

11 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{361d263e-0ee1-47e9-8fc2-0f127f1c2d7e-24_553_951_258_557} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 represents the plan view of part of a horizontal floor, where $A B$ and $C D$ represent fixed vertical walls, with $A B$ parallel to $C D$. A small ball is projected along the floor towards wall $A B$. Immediately before hitting wall $A B$, the ball is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\alpha$ to $A B$, where $0 < \alpha < \frac { \pi } { 2 }$ The ball hits wall $A B$ and then hits wall $C D$.
After the impact with wall $C D$, the ball is moving at angle $\frac { 1 } { 2 } \alpha$ to $C D$.
The coefficient of restitution between the ball and wall $A B$ is $\frac { 2 } { 3 }$
The coefficient of restitution between the ball and wall $C D$ is also $\frac { 2 } { 3 }$
The floor and the walls are modelled as being smooth. The ball is modelled as a particle.

Show that $\tan \left( \frac { 1 } { 2 } \alpha \right) = \frac { 1 } { 3 }$
Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.

Edexcel FM1 2021 June Q1

9 marks Standard +0.3

A van of mass 900 kg is moving along a straight horizontal road.

At the instant when the speed of the van is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resistance to the motion of the van is modelled as a force of magnitude $( 500 + 7 v ) \mathrm { N }$. When the engine of the van is working at a constant rate of 18 kW , the van is moving along the road at a constant speed $V \mathrm {~ms} ^ { - 1 }$

Find the value of $V$. Later on, the van is moving up a straight road that is inclined to the horizontal at an angle $\theta$, where $\sin \theta = \frac { 1 } { 21 }$ At the instant when the speed of the van is $v \mathrm {~ms} ^ { - 1 }$, the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude $( 500 + 7 v ) \mathrm { N }$. The engine of the van is again working at a constant rate of 18 kW .
Find the acceleration of the van at the instant when $v = 15$

Edexcel FM1 2021 June Q2

14 marks Standard +0.3

Two particles, $A$ and $B$, are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.

Particle $A$ has mass $5 m$ and particle $B$ has mass $3 m$.
The coefficient of restitution between $A$ and $B$ is $e$, where $e > 0$
Immediately after the collision the speed of $A$ is $v$ and the speed of $B$ is $2 v$.
Given that $A$ and $B$ are moving in the same direction after the collision,

find the set of possible values of $e$. Given also that the kinetic energy of $A$ immediately after the collision is $16 \%$ of the kinetic energy of $A$ immediately before the collision,
find
1. the value of $e$,
2. the magnitude of the impulse received by $A$ in the collision, giving your answer in terms of $m$ and $v$.

Edexcel FM1 2021 June Q3

14 marks Challenging +1.2

\hspace{0pt} [In this question, $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere $P$ has mass 0.3 kg . Another smooth uniform sphere $Q$, with the same radius as $P$, has mass 0.5 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of $P$ is $( u \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$, where $u$ is a positive constant, and the velocity of $Q$ is $( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ At the instant when the spheres collide, the line joining their centres is parallel to $\mathbf { i }$.
The coefficient of restitution between $P$ and $Q$ is $\frac { 3 } { 5 }$
As a result of the collision, the direction of motion of $P$ is deflected through an angle of $90 ^ { \circ }$ and the direction of motion of $Q$ is deflected through an angle of $\alpha ^ { \circ }$

Find the value of $u$
Find the value of $\alpha$
State how you have used the fact that $P$ and $Q$ have equal radii.

Edexcel FM1 2021 June Q4

8 marks Standard +0.3

A particle $P$ has mass 0.5 kg . It is moving in the $x y$ plane with velocity $8 \mathbf { i } \mathrm {~ms} ^ { - 1 }$ when it receives an impulse $\lambda ( - \mathbf { i } + \mathbf { j } )$ Ns, where $\lambda$ is a positive constant.

The angle between the direction of motion of $P$ immediately before receiving the impulse and the direction of motion of $P$ immediately after receiving the impulse is $\theta ^ { \circ }$ Immediately after receiving the impulse, $P$ is moving with speed $4 \sqrt { 10 } \mathrm {~ms} ^ { - 1 }$
Find (i) the value of $\lambda$
(ii) the value of $\theta$

Edexcel FM1 2021 June Q5

10 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7f077b82-6b39-4cb5-8574-bfa308c88df3-16_575_665_246_699} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 represents the plan view of part of a horizontal floor, where $A B$ and $B C$ represent fixed vertical walls, with $A B$ perpendicular to $B C$. A small ball is projected along the floor towards the wall $A B$. Immediately before hitting the wall $A B$ the ball is moving with speed $v \mathrm {~ms} ^ { - 1 }$ at an angle $\theta$ to $A B$. The ball hits the wall $A B$ and then hits the wall $B C$.
The coefficient of restitution between the ball and the wall $A B$ is $\frac { 1 } { 3 }$
The coefficient of restitution between the ball and the wall $B C$ is $e$.
The floor and the walls are modelled as being smooth.
The ball is modelled as a particle.
The ball loses half of its kinetic energy in the impact with the wall $A B$.

Find the exact value of $\cos \theta$. The ball loses half of its remaining kinetic energy in the impact with the wall $B C$.
Find the exact value of $e$.

Edexcel FM1 2021 June Q6

11 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7f077b82-6b39-4cb5-8574-bfa308c88df3-20_401_814_246_628} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A light elastic spring has natural length $3 l$ and modulus of elasticity $3 m g$.
One end of the spring is attached to a fixed point $X$ on a rough inclined plane.
The other end of the spring is attached to a package $P$ of mass $m$.
The plane is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha = \frac { 3 } { 4 }$
The package is initially held at the point $Y$ on the plane, where $X Y = l$. The point $Y$ is higher than $X$ and $X Y$ is a line of greatest slope of the plane, as shown in Figure 2. The package is released from rest at $Y$ and moves up the plane.
The coefficient of friction between $P$ and the plane is $\frac { 1 } { 3 }$
By modelling $P$ as a particle,

show that the acceleration of $P$ at the instant when $P$ is released from rest is $\frac { 17 } { 15 } \mathrm {~g}$
find, in terms of $g$ and $l$, the speed of $P$ at the instant when the spring first reaches its natural length of 31 .

Edexcel FM1 2021 June Q7

9 marks Standard +0.8

\hspace{0pt} [In this question, $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in a horizontal plane.]

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7f077b82-6b39-4cb5-8574-bfa308c88df3-24_543_789_294_639} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 represents the plan view of part of a smooth horizontal floor, where $A B$ is a fixed smooth vertical wall. The direction of $\overrightarrow { A B }$ is in the direction of the vector $( \mathbf { i } + \mathbf { j } )$
A small ball of mass 0.25 kg is moving on the floor when it strikes the wall $A B$.
Immediately before its impact with the wall $A B$, the velocity of the ball is $( 8 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
Immediately after its impact with the wall $A B$, the velocity of the ball is $\mathbf { v m s } ^ { - 1 }$
The coefficient of restitution between the ball and the wall is $\frac { 1 } { 3 }$
By modelling the ball as a particle,

show that $\mathbf { v } = 4 \mathbf { i } + 6 \mathbf { j }$
Find the magnitude of the impulse received by the ball in the impact.

Edexcel FM1 2022 June Q1

8 marks Standard +0.3

A particle $A$ of mass $3 m$ and a particle $B$ of mass $m$ are moving along the same straight line on a smooth horizontal surface. The particles are moving in opposite directions towards each other when they collide directly.

Immediately before the collision, the speed of $A$ is $k u$ and the speed of $B$ is $u$. Immediately after the collision, the speed of $A$ is $v$ and the speed of $B$ is $2 v$. The magnitude of the impulse received by $B$ in the collision is $\frac { 3 } { 2 } m u$.

Find $v$ in terms of $u$ only.
Find the two possible values of $k$.

Edexcel FM1 2022 June Q2

8 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-06_287_846_246_612} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A van of mass 600 kg is moving up a straight road which is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 15 }$. The van is towing a trailer of mass 150 kg . The van is attached to the trailer by a towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 200 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 100 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 12 kW .
Find the tension in the towbar at the instant when the speed of the van is $9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Edexcel FM1 2022 June Q3

5 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-10_302_442_244_813} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 0.5 kg is moving in a straight line with speed $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it receives an impulse of magnitude 3 Ns .
The angle between the direction of motion of $P$ immediately before receiving the impulse and the line of action of the impulse is $\alpha$, where $\tan \alpha = \frac { 4 } { 3 }$, as shown in Figure 2. Find the speed of $P$ immediately after receiving the impulse.

Edexcel FM1 2022 June Q4

9 marks Challenging +1.2

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-12_387_929_246_568} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two smooth uniform spheres, $A$ and $B$, have equal radii. The mass of $A$ is $3 m$ and the mass of $B$ is $4 m$. The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, $A$ is moving with speed $3 u$ at $30 ^ { \circ }$ to the line of centres of the spheres and $B$ is moving with speed $2 u$ at $30 ^ { \circ }$ to the line of centres of the spheres. The direction of motion of $B$ is turned through an angle of $90 ^ { \circ }$ by the collision, as shown in Figure 3.

Find the size of the angle through which the direction of motion of $A$ is turned as a result of the collision.
Find, in terms of $m$ and $u$, the magnitude of the impulse received by $B$ in the collision.

Edexcel FM1 2022 June Q5

10 marks Standard +0.8

Two particles, $P$ and $Q$, are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
The mass of $P$ is $3 m$ and the mass of $Q$ is $4 m$.
Immediately before the collision the speed of $P$ is $2 u$ and the speed of $Q$ is $u$.
The coefficient of restitution between $P$ and $Q$ is $e$.
1. Show that the speed of $Q$ immediately after the collision is $\frac { u } { 7 } ( 9 e + 2 )$
After the collision with $P$, particle $Q$ collides directly with a fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of $Q$.
The coefficient of restitution between $Q$ and the wall is $\frac { 1 } { 2 }$
Find the complete range of possible values of $e$ for which there is a second collision between $P$ and $Q$.

Edexcel FM1 2022 June Q6

13 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_497_1337_246_365} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Two blocks, $A$ and $B$, of masses 2 kg and 4 kg respectively are attached to the ends of a light inextensible string. Initially $A$ is held on a fixed rough plane. The plane is inclined to horizontal ground at an angle $\theta$, where $\tan \theta = \frac { 3 } { 4 }$
The string passes over a small smooth light pulley $P$ that is fixed at the top of the plane. The part of the string from $A$ to $P$ is parallel to a line of greatest slope of the plane. Block $A$ is held on the plane with the distance $A P$ greater than 3 m .
Block $B$ hangs freely below $P$ at a distance of 3 m above the ground, as shown in Figure 4. The coefficient of friction between $A$ and the plane is $\mu$
Block $A$ is released from rest with the string taut.
By modelling the blocks as particles,

find the potential energy lost by the whole system as a result of $B$ falling 3 m . Given that the speed of $B$ at the instant it hits the ground is $4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and ignoring air resistance,
use the work-energy principle to find the value of $\mu$ After $B$ hits the ground, $A$ continues to move up the plane but does not reach the pulley in the subsequent motion.
Block $A$ comes to instantaneous rest after moving a total distance of ( $3 + d$ ) m from its point of release. Ignoring air resistance,
use the work-energy principle to find the value of $d$
\includegraphics[max width=\textwidth, alt={}, center]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_2255_50_309_1981}

Edexcel FM1 2022 June Q7

12 marks Standard +0.8

A spring of natural length $a$ has one end attached to a fixed point $A$. The other end of the spring is attached to a package $P$ of mass $m$.
The package $P$ is held at rest at the point $B$, which is vertically below $A$ such that $A B = 3 a$.
After being released from rest at $B$, the package $P$ first comes to instantaneous rest at $A$. Air resistance is modelled as being negligible.

By modelling the spring as being light and modelling $P$ as a particle,

show that the modulus of elasticity of the spring is $2 m g$
1. Show that $P$ attains its maximum speed when the extension of the spring is $\frac { 1 } { 2 } a$
2. Use the principle of conservation of mechanical energy to find the maximum speed, giving your answer in terms of $a$ and $g$. In reality, the spring is not light.
State one way in which this would affect your energy equation in part (b).

Edexcel FM1 2022 June Q8

10 marks Challenging +1.2

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-28_567_1406_244_333} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 represents the plan view of part of a smooth horizontal floor, where $R S$ and $S T$ are smooth fixed vertical walls. The vector $\overrightarrow { R S }$ is in the direction of $\mathbf { i }$ and the vector $\overrightarrow { S T }$ is in the direction of $( 2 \mathbf { i } + \mathbf { j } )$. A small ball $B$ is projected across the floor towards $R S$. Immediately before the impact with $R S$, the velocity of $B$ is $( 6 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. The ball bounces off $R S$ and then hits $S T$. The ball is modelled as a particle.
Given that the coefficient of restitution between $B$ and $R S$ is $e$,

find the full range of possible values of $e$. It is now given that $e = \frac { 1 } { 4 }$ and that the coefficient of restitution between $B$ and $S T$ is $\frac { 1 } { 2 }$
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $B$ immediately after its impact with $S T$.

Edexcel FM1 2023 June Q1

6 marks Standard +0.3

A particle $P$ of mass 2 kg is moving with velocity $( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse $( - 6 \mathbf { i } + 42 \mathbf { j } )$ N s.
1. Find the speed of $P$ immediately after receiving the impulse.
The angle through which the direction of motion of $P$ has been deflected by the impulse is $\alpha ^ { \circ }$
Find the value of $\alpha$

Edexcel FM1 2023 June Q2

8 marks Standard +0.3

A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW .

Find the value of $U$. The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N . The resistance to the motion of the trailer is a constant force of magnitude 300 N . The engine of the car is working at a constant rate of 16 kW .
The tow rope is modelled as being light and inextensible.
Using the model,
find the tension in the tow rope at the instant when the speed of the car is $\frac { 20 } { 3 } \mathrm {~ms} ^ { - 1 }$

Edexcel FM1 2023 June Q3

10 marks Standard +0.8

A particle $P$ of mass $2 m$ is moving in a straight line with speed $3 u$ on a smooth horizontal plane. It collides directly with a particle $Q$ of mass $m$ that is moving on the plane with speed $2 u$ in the opposite direction to $P$.
The coefficient of restitution between $P$ and $Q$ is $e$, where $e > \frac { 4 } { 5 }$
1. Show that the speed of $Q$ immediately after the collision is $\frac { ( 4 + 10 e ) u } { 3 }$
After the collision $Q$ hits a smooth fixed vertical wall that is perpendicular to the direction of motion of $Q$. The coefficient of restitution between $Q$ and the wall is $f$.
Find, in terms of $\boldsymbol { e }$, the set of values of $f$ for which there will be a second collision between $P$ and $Q$.

Edexcel FM1 2023 June Q4

15 marks Standard +0.8

A light elastic string has natural length $2 a$ and modulus of elasticity $4 m g$. One end of the elastic string is attached to a fixed point $O$. A particle $P$ of mass $m$ is attached to the other end of the elastic string.
The particle $P$ hangs freely in equilibrium at the point $E$, which is vertically below $O$
1. Find the length $O E$.
Particle $P$ is now pulled vertically downwards to the point $A$, where $O A = 4 a$, and released from rest. The resistance to the motion of $P$ is a constant force of magnitude $\frac { 1 } { 4 } m g$.
Find, in terms of $a$ and $g$, the speed of $P$ after it has moved a distance $a$. Particle $P$ is now held at $O$ Particle $P$ is released from rest and reaches its maximum speed at the point $B$. The resistance to the motion of $P$ is again a constant force of magnitude $\frac { 1 } { 4 } m g$.
Find the distance $O B$.

Edexcel FM1 2023 June Q5

10 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0da9cd5b-6f6f-4607-bd4f-c8ae164466ae-16_758_1399_280_333} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A smooth uniform sphere $S$ of mass $m$ is moving with speed $U$ on a smooth horizontal plane. The sphere $S$ collides obliquely with another uniform sphere of mass $M$ which is at rest on the plane. The two spheres have the same radius. Immediately before the collision the direction of motion of $S$ makes an angle $\alpha$, where $0 < \alpha < 90 ^ { \circ }$, with the line joining the centres of the spheres. Immediately after the collision the direction of motion of $S$ makes an angle $\beta$ with the line joining the centres of the spheres, as shown in Figure 1. The coefficient of restitution between the spheres is $e$.

Show that $\tan \beta = \frac { ( m + M ) \tan \alpha } { ( m - e M ) }$ Given that $m = e M$,
show that the directions of motion of the two spheres immediately after the collision are perpendicular.

Edexcel FM1 2023 June Q6

12 marks Challenging +1.2

A particle $P$ of mass $m$ is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle $\alpha$, where $0 < \alpha \leqslant 45 ^ { \circ }$

At the instant immediately before the impact, the speed of $P$ is $u$.
At the instant immediately after the impact, $P$ is moving horizontally with speed $v$.

Show that the magnitude of the impulse exerted on the plane by $P$ is $m u \sec \alpha$ The coefficient of restitution between $P$ and the plane is $e$, where $e > 0$
Show that $v ^ { 2 } = u ^ { 2 } \left( \sin ^ { 2 } \alpha + e ^ { 2 } \cos ^ { 2 } \alpha \right)$
Show that the kinetic energy lost by $P$ in the impact is $$\frac { 1 } { 2 } m u ^ { 2 } \left( 1 - e ^ { 2 } \right) \cos ^ { 2 } \alpha$$
Hence find, in terms of $m$, $u$ and $e$ only, the kinetic energy lost by $P$ in the impact.

Edexcel FM1 2023 June Q7

14 marks

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0da9cd5b-6f6f-4607-bd4f-c8ae164466ae-24_721_1367_280_349} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A small smooth snooker ball is projected from the corner $A$ of a horizontal rectangular snooker table $A B C D$. The ball is projected so it first hits the side $D C$ at the point $P$, then hits the side $C B$ at the point $Q$ and then returns to $A$. Angle $A P D = \alpha$, Angle $Q P C = \beta$, Angle $A Q B = \gamma$
The ball moves along $A P$ with speed $U$, along $P Q$ with speed $V$ and along $Q A$ with speed $W$, as shown in Figure 2. The coefficient of restitution between the ball and side $D C$ is $e _ { 1 }$
The coefficient of restitution between the ball and side $C B$ is $e _ { 2 }$
The ball is modelled as a particle. \section*{Use the model to answer all parts of this question.}

Show that $\tan \beta = e _ { 1 } \tan \alpha$
Hence show that $e _ { 1 } \tan \alpha = e _ { 2 } \cot \gamma$
By considering (angle $A P Q$ + angle $A Q P$ ) or otherwise, show that it would be possible for the ball to return to $A$ only if $e _ { 2 } > e _ { 1 }$ If instead $e _ { 1 } = e _ { 2 }$, the ball would not return to $A$.
Given that $e _ { 1 } = e _ { 2 }$
use the result from part (b) to describe the path of the ball after it hits $C B$ at $Q$, explaining your answer.

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