Questions M3 - A-Level Maths

Given that $\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$, show that the speed of $B$ immediately after the collision is $$\frac { \sqrt { 3 } } { 4 } u ( 1 + e )$$
Find, in terms of $u$ and $e$, the components of the velocity of $A$, parallel and perpendicular to the line of centres, immediately after the collision.
Given that $e = \frac { 2 } { 3 }$, find the angle that the velocity of $A$ makes with the line of centres immediately after the collision. Give your answer to the nearest degree.
(3 marks)

AQA M3 2007 June Q7

7 A particle is projected from a point on a plane which is inclined at an angle $\alpha$ to the horizontal. The particle is projected up the plane with velocity $u$ at an angle $\theta$ above the plane. The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane.
\includegraphics[max width=\textwidth, alt={}, center]{daea0765-041a-4569-a535-f90fe4708313-5_401_748_516_644}

Using the identity $\cos ( A + B ) = \cos A \cos B - \sin A \sin B$, show that the range up the plane is $$\frac { 2 u ^ { 2 } \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
Hence, using the identity $2 \sin A \cos B = \sin ( A + B ) + \sin ( A - B )$, show that, as $\theta$ varies, the range up the plane is a maximum when $\theta = \frac { \pi } { 4 } - \frac { \alpha } { 2 }$.
Given that the particle strikes the plane at right angles, show that $$2 \tan \theta = \cot \alpha$$

AQA M3 2008 June Q1

1 The speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of a wave travelling along the surface of a sea is believed to depend on
the depth of the sea, $d \mathrm {~m}$,
the density of the water, $\rho \mathrm { kg } \mathrm { m } ^ { - 3 }$,
the acceleration due to gravity, $g$, and
a dimensionless constant, $k$
so that $$v = k d ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }$$ where $\alpha , \beta$ and $\gamma$ are constants.
By using dimensional analysis, show that $\beta = 0$ and find the values of $\alpha$ and $\gamma$.

AQA M3 2008 June Q2

2 The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed due east and due north respectively.
Two runners, Albina and Brian, are running on level parkland with constant velocities of $( 5 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ respectively. Initially, the position vectors of Albina and Brian are $( - 60 \mathbf { i } + 160 \mathbf { j } ) \mathrm { m }$ and $( 40 \mathbf { i } - 90 \mathbf { j } ) \mathrm { m }$ respectively, relative to a fixed origin in the parkland.

Write down the velocity of Brian relative to Albina.
Find the position vector of Brian relative to Albina $t$ seconds after they leave their initial positions.
Hence determine whether Albina and Brian will collide if they continue running with the same velocities.

AQA M3 2008 June Q3

3 A particle of mass 0.2 kg lies at rest on a smooth horizontal table. A horizontal force of magnitude $F$ newtons acts on the particle in a constant direction for 0.1 seconds. At time $t$ seconds, $$F = 5 \times 10 ^ { 3 } t ^ { 2 } , \quad 0 \leqslant t \leqslant 0.1$$ Find the value of $t$ when the speed of the particle is $2 \mathrm {~ms} ^ { - 1 }$.
(4 marks)

AQA M3 2008 June Q4

4 Two smooth spheres, $A$ and $B$, have equal radii and masses $m$ and $2 m$ respectively. The spheres are moving on a smooth horizontal plane. The sphere $A$ has velocity ( $4 \mathbf { i } + 3 \mathbf { j }$ ) when it collides with the sphere $B$ which has velocity $( - 2 \mathbf { i } + 2 \mathbf { j } )$. After the collision, the velocity of $B$ is $( \mathbf { i } + \mathbf { j } )$.

Find the velocity of $A$ immediately after the collision.
Find the angle between the velocities of $A$ and $B$ immediately after the collision.
Find the impulse exerted by $B$ on $A$.
State, as a vector, the direction of the line of centres of $A$ and $B$ when they collide.
(1 mark)

AQA M3 2008 June Q5

5 A boy throws a small ball from a height of 1.5 m above horizontal ground with initial velocity $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\alpha$ above the horizontal. The ball hits a small can placed on a vertical wall of height 2.5 m , which is at a horizontal distance of 5 m from the initial position of the ball, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-3_499_1180_1283_424}

Show that $\alpha$ satisfies the equation $$49 \tan ^ { 2 } \alpha - 200 \tan \alpha + 89 = 0$$
Find the two possible values of $\alpha$, giving your answers to the nearest $0.1 ^ { \circ }$.
1. To knock the can off the wall, the horizontal component of the velocity of the ball must be greater than $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Show that, for one of the possible values of $\alpha$ found in part (b), the can will be knocked off the wall, and for the other, it will not be knocked off the wall.
  (3 marks)
2. Given that the can is knocked off the wall, find the direction in which the ball is moving as it hits the can.

AQA M3 2008 June Q6

6 A small smooth ball of mass $m$, moving on a smooth horizontal surface, hits a smooth vertical wall and rebounds. The coefficient of restitution between the wall and the ball is $\frac { 3 } { 4 }$. Immediately before the collision, the ball has velocity $u$ and the angle between the ball's direction of motion and the wall is $\alpha$. The ball's direction of motion immediately after the collision is at right angles to its direction of motion before the collision, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-4_483_344_657_854}

Show that $\tan \alpha = \frac { 2 } { \sqrt { 3 } }$.
Find, in terms of $u$, the speed of the ball immediately after the collision.
The force exerted on the ball by the wall acts for 0.1 seconds. Given that $m = 0.2 \mathrm {~kg}$ and $u = 4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the average force exerted by the wall on the ball.

AQA M3 2008 June Q7

7 A projectile is fired with speed $u$ from a point $O$ on a plane which is inclined at an angle $\alpha$ to the horizontal. The projectile is fired at an angle $\theta$ to the inclined plane and moves in a vertical plane through a line of greatest slope of the inclined plane. The projectile lands at a point $P$, lower down the inclined plane, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-5_415_1098_495_463}

Find, in terms of $u , g , \theta$ and $\alpha$, the greatest perpendicular distance of the projectile from the plane.
1. Find, in terms of $u , g , \theta$ and $\alpha$, the time of flight from $O$ to $P$.
2. By using the identity $\cos A \cos B + \sin A \sin B = \cos ( A - B )$, show that the distance $O P$ is given by $\frac { 2 u ^ { 2 } \sin \theta \cos ( \theta - \alpha ) } { g \cos ^ { 2 } \alpha }$.
3. Hence, by using the identity $2 \sin A \cos B = \sin ( A + B ) + \sin ( A - B )$ or otherwise, show that, as $\theta$ varies, the maximum possible distance $O P$ is $\frac { u ^ { 2 } } { g ( 1 - \sin \alpha ) }$.
  (5 marks)

AQA M3 2010 June Q1

1 A tank containing a liquid has a small hole in the bottom through which the liquid escapes. The speed, $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at which the liquid escapes is given by $$u = C V \rho g$$ where $V \mathrm {~m} ^ { 3 }$ is the volume of the liquid in the tank, $\rho \mathrm { kg } \mathrm { m } ^ { - 3 }$ is the density of the liquid, $g$ is the acceleration due to gravity and $C$ is a constant. By using dimensional analysis, find the dimensions of $C$.
□
□

\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-03_2484_1709_223_153}

AQA M3 2010 June Q2

2 A projectile is fired from a point $O$ on top of a hill with initial velocity $80 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\theta$ above the horizontal and moves in a vertical plane. The horizontal and upward vertical distances of the projectile from $O$ are $x$ metres and $y$ metres respectively.

1. Show that, during the flight, the equation of the trajectory of the projectile is given by $$y = x \tan \theta - \frac { g x ^ { 2 } } { 12800 } \left( 1 + \tan ^ { 2 } \theta \right)$$
2. The projectile hits a target $A$, which is 20 m vertically below $O$ and 400 m horizontally from $O$.
  \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-04_392_1031_970_460} Show that $$49 \tan ^ { 2 } \theta - 160 \tan \theta + 41 = 0$$
1. Find the two possible values of $\theta$. Give your answers to the nearest $0.1 ^ { \circ }$.
2. Hence find the shortest possible time of the flight of the projectile from $O$ to $A$.
State a necessary modelling assumption for answering part (a)(i).
\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-05_2484_1709_223_153}

\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-07_2484_1709_223_153}

AQA M3 2010 June Q3

3 Three smooth spheres, $A , B$ and $C$, of equal radii have masses $1 \mathrm {~kg} , 3 \mathrm {~kg}$ and $x \mathrm {~kg}$ respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with $B$ between $A$ and $C$. The sphere $A$ is projected with speed $3 u$ directly towards $B$ and collides with it.
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-08_250_835_511_605} The coefficient of restitution between each pair of spheres is $\frac { 1 } { 3 }$.

Show that $A$ is brought to rest by the impact and find the speed of $B$ immediately after the collision in terms of $u$.
Subsequently, $B$ collides with $C$. Show that the speed of $C$ immediately after the collision is $\frac { 4 u } { 3 + x }$.
Find the speed of $B$ immediately after the collision in terms of $u$ and $x$.
Show that $B$ will collide with $A$ again if $x > 9$.
Given that $x = 5$, find the magnitude of the impulse exerted on $C$ by $B$ in terms of $u$.
\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-09_2484_1709_223_153}

\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-10_2484_1712_223_153}

\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-11_2484_1709_223_153}

AQA M3 2010 June Q4

4 The unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are directed east, north and vertically upwards respectively. At time $t = 0$, the position vectors of two small aeroplanes, $A$ and $B$, relative to a fixed origin $O$ are $( - 60 \mathbf { i } + 30 \mathbf { k } ) \mathrm { km }$ and $( - 40 \mathbf { i } + 10 \mathbf { j } - 10 \mathbf { k } ) \mathrm { km }$ respectively. The aeroplane $A$ is flying with constant velocity $( 250 \mathbf { i } + 50 \mathbf { j } - 100 \mathbf { k } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and the aeroplane $B$ is flying with constant velocity $( 200 \mathbf { i } + 25 \mathbf { j } + 50 \mathbf { k } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.

Write down the position vectors of $A$ and $B$ at time $t$ hours.
Show that the position vector of $A$ relative to $B$ at time $t$ hours is $( ( - 20 + 50 t ) \mathbf { i } + ( - 10 + 25 t ) \mathbf { j } + ( 40 - 150 t ) \mathbf { k } ) \mathrm { km }$.
Show that $A$ and $B$ do not collide.
Find the value of $t$ when $A$ and $B$ are closest together.
\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-13_2484_1709_223_153}

AQA M3 2010 June Q5

5 A smooth sphere is moving on a smooth horizontal surface when it strikes a smooth vertical wall and rebounds. Immediately before the impact, the sphere is moving with speed $4 \mathrm {~ms} ^ { - 1 }$ and the angle between the sphere's direction of motion and the wall is $\alpha$. Immediately after the impact, the sphere is moving with speed $v \mathrm {~ms} ^ { - 1 }$ and the angle between the sphere's direction of motion and the wall is $40 ^ { \circ }$. The coefficient of restitution between the sphere and the wall is $\frac { 2 } { 3 }$.
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-14_529_250_831_909}

Show that $\tan \alpha = \frac { 3 } { 2 } \tan 40 ^ { \circ }$.
Find the value of $v$.
\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-15_2484_1709_223_153}

AQA M3 2010 June Q6

6 Two smooth spheres, $A$ and $B$, have equal radii and masses 1 kg and 2 kg respectively. The sphere $A$ is moving with velocity $( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and the sphere $B$ is moving with velocity $( - \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ on the same smooth horizontal surface. The spheres collide when their line of centres is parallel to the unit vector $\mathbf { i }$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-16_456_1052_721_550}

Briefly state why the components of the velocities of $A$ and $B$ parallel to the unit vector $\mathbf { j }$ are not changed by the collision.
The coefficient of restitution between the spheres is 0.5 . Find the velocities of $A$ and $B$ immediately after the collision. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-17_2484_1709_223_153}
$7 \quad$ A ball is projected from a point $O$ on a smooth plane which is inclined at an angle of $35 ^ { \circ }$ above the horizontal. The ball is projected with velocity $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ above the plane, as shown in the diagram. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane at the point $A$.
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-18_321_838_605_577}
Find the components of the velocity of the ball, parallel and perpendicular to the plane, as it strikes the inclined plane at $A$.
On striking the plane at $A$, the ball rebounds. The coefficient of restitution between the plane and the ball is $\frac { 4 } { 5 }$. Show that the ball next strikes the plane at a point lower down than $A$.
\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-19_2484_1709_223_153}

AQA M3 2011 June Q1

1 A ball of mass 0.2 kg is hit directly by a bat. Just before the impact, the ball is travelling horizontally with speed $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Just after the impact, the ball is travelling horizontally with speed $32 \mathrm {~ms} ^ { - 1 }$ in the opposite direction.

Find the magnitude of the impulse exerted on the ball.
At time $t$ seconds after the ball first comes into contact with the bat, the force exerted by the bat on the ball is $k \left( 0.9 t - 10 t ^ { 2 } \right)$ newtons, where $k$ is a constant and $0 \leqslant t \leqslant 0.09$. The bat stays in contact with the ball for 0.09 seconds. Find the value of $k$.

AQA M3 2011 June Q2

2 The time, $t$, for a single vibration of a piece of taut string is believed to depend on
the length of the taut string, $l$,
the tension in the string, $F$,
the mass per unit length of the string, $q$, and
a dimensionless constant, $k$,
such that $$t = k l ^ { \alpha } F ^ { \beta } q ^ { \gamma }$$ where $\alpha , \beta$ and $\gamma$ are constants.
By using dimensional analysis, find the values of $\alpha , \beta$ and $\gamma$.

AQA M3 2011 June Q3

3 (In this question, use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.)
A golf ball is hit from a point $O$ on a horizontal golf course with a velocity of $40 \mathrm {~ms} ^ { - 1 }$ at an angle of elevation $\theta$. The golf ball travels in a vertical plane through $O$. During its flight, the horizontal and upward vertical distances of the golf ball from $O$ are $x$ and $y$ metres respectively.

Show that the equation of the trajectory of the golf ball during its flight is given by $$x ^ { 2 } \tan ^ { 2 } \theta - 320 x \tan \theta + \left( x ^ { 2 } + 320 y \right) = 0$$
1. The golf ball hits the top of a tree, which has a vertical height of 8 m and is at a horizontal distance of 150 m from $O$. Find the two possible values of $\theta$.
2. Which value of $\theta$ gives the shortest possible time for the golf ball to travel from $O$ to the top of the tree? Give a reason for your choice of $\theta$.

AQA M3 2011 June Q4

4 The unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are directed due east, due north and vertically upwards respectively. A helicopter, $A$, is travelling in the direction of the vector $- 2 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }$ with constant speed $140 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Another helicopter, $B$, is travelling in the direction of the vector $2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$ with constant speed $60 \mathrm {~km} \mathrm {~h} ^ { - 1 }$.

Find the velocity of $A$ relative to $B$.
Initially, the position vectors of $A$ and $B$ are $( 4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }$ and $( - 3 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }$ respectively, relative to a fixed origin. Write down the position vector of $A$ relative to $B , t$ hours after they leave their initial positions.
Find the distance between $A$ and $B$ when they are closest together.

\includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-10_2486_1714_221_153}

\includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-11_2486_1714_221_153}

AQA M3 2011 June Q5

5 A ball is dropped from a height of 2.5 m above a horizontal floor. The ball bounces repeatedly on the floor.

Find the speed of the ball when it first hits the floor.
The coefficient of restitution between the ball and the floor is $e$.
1. Show that the time taken between the first contact of the ball with the floor and the second contact of the ball with the floor is $\frac { 10 e } { 7 }$ seconds.
2. Find, in terms of $e$, the time taken between the second contact and the third contact of the ball with the floor.
Find, in terms of $e$, the total vertical distance travelled by the ball from when it is dropped until its third contact with the floor.
State a modelling assumption for answering this question, other than the ball being a particle.

AQA M3 2011 June Q6

6 A projectile is fired from a point $O$ on a plane which is inclined at an angle of $20 ^ { \circ }$ to the horizontal. The projectile is fired up the plane with velocity $u \mathrm {~ms} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ to the inclined plane. The projectile travels in a vertical plane containing a line of greatest slope of the inclined plane. The projectile hits a target $T$ on the inclined plane.
\includegraphics[max width=\textwidth, alt={}, center]{0590950d-145c-4ae2-bc3c-f61a9433d158-16_481_922_664_593}

Given that $O T = 200 \mathrm {~m}$, determine the value of $u$.
Find the greatest perpendicular distance of the projectile from the inclined plane.
(4 marks)

\includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-18_2486_1714_221_153}

\includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-19_2486_1714_221_153}

AQA M3 2011 June Q7

7 Two smooth spheres, $A$ and $B$, have equal radii and masses $4 m$ and $3 m$ respectively. The sphere $A$ is moving on a smooth horizontal surface and collides with the sphere $B$, which is stationary on the same surface. Just before the collision, $A$ is moving with speed $u$ at an angle of $30 ^ { \circ }$ to the line of centres, as shown in the diagram below. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Before collision} \includegraphics[alt={},max width=\textwidth]{0590950d-145c-4ae2-bc3c-f61a9433d158-20_362_933_664_450}

\end{figure} Immediately after the collision, the direction of motion of $A$ makes an angle $\alpha$ with the line of centres, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{0590950d-145c-4ae2-bc3c-f61a9433d158-20_449_927_1244_456} The coefficient of restitution between the spheres is $\frac { 5 } { 9 }$.

Find the value of $\alpha$.
Find, in terms of $m$ and $u$, the magnitude of the impulse exerted on $B$ during the collision.
\includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-23_2349_1707_221_153}

AQA M3 2012 June Q1

1 An ice-hockey player has mass 60 kg . He slides in a straight line at a constant speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on the horizontal smooth surface of an ice rink towards the vertical perimeter wall of the rink, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-02_476_594_769_715} The player collides directly with the wall, and remains in contact with the wall for 0.5 seconds. At time $t$ seconds after coming into contact with the wall, the force exerted by the wall on the player is $4 \times 10 ^ { 4 } t ^ { 2 } ( 1 - 2 t )$ newtons, where $0 \leqslant t \leqslant 0.5$.

Find the magnitude of the impulse exerted by the wall on the player.
The player rebounds from the wall. Find the player's speed immediately after the collision.

AQA M3 2012 June Q2

2 A pile driver of mass $m _ { 1 }$ falls from a height $h$ onto a pile of mass $m _ { 2 }$, driving the pile a distance $s$ into the ground. The pile driver remains in contact with the pile after the impact. A resistance force $R$ opposes the motion of the pile into the ground. Elizabeth finds an expression for $R$ as $$R = \frac { g } { s } \left[ s \left( m _ { 1 } + m _ { 2 } \right) + \frac { h \left( m _ { 1 } \right) ^ { 2 } } { m _ { 1 } + m _ { 2 } } \right]$$ where $g$ is the acceleration due to gravity.
Determine whether the expression is dimensionally consistent.

AQA M3 2012 June Q3

3 (In this question, take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.)
A projectile is fired from a point $O$ with speed $u$ at an angle of elevation $\alpha$ above the horizontal so as to pass through a point $P$. The projectile travels in a vertical plane through $O$ and $P$. The point $P$ is at a horizontal distance $2 k$ from $O$ and at a vertical distance $k$ above $O$.

Show that $\alpha$ satisfies the equation $$20 k \tan ^ { 2 } \alpha - 2 u ^ { 2 } \tan \alpha + u ^ { 2 } + 20 k = 0$$
Deduce that $$u ^ { 4 } - 20 k u ^ { 2 } - 400 k ^ { 2 } \geqslant 0$$

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