Questions - AQA M3 - A-Level Maths

Given that $v _ { A } = 200$ :
1. find a bearing, to one decimal place, on which $A$ could travel in order to intercept $H$;
2. find the time, in minutes, that it would take $A$ to intercept $H$ on this bearing.
Given that $v _ { A } = 150$, find the bearing on which $A$ should travel in order to approach $H$ as closely as possible. Give your answer to one decimal place.
\includegraphics[max width=\textwidth, alt={}]{3a1726d9-1b0c-41de-8b43-56019e18aac1-20_2253_1691_221_153}

AQA M3 2014 June Q2

6 marks

2 A rod, of length $x \mathrm {~m}$ and moment of inertia $I \mathrm {~kg} \mathrm {~m} ^ { 2 }$, is free to rotate in a vertical plane about a fixed smooth horizontal axis through one end. When the rod is hanging at rest, its lower end receives an impulse of magnitude $J$ Ns, which is just sufficient for the rod to complete full revolutions. It is thought that there is a relationship between $J , x , I$, the acceleration due to gravity $g \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and a dimensionless constant $k$, such that $$J = k x ^ { \alpha } I ^ { \beta } g ^ { \gamma }$$ where $\alpha , \beta$ and $\gamma$ are constants.
Find the values of $\alpha , \beta$ and $\gamma$ for which this relationship is dimensionally consistent.
[0pt] [6 marks]

AQA M3 2014 June Q3

3 A particle of mass 0.5 kg is moving in a straight line on a smooth horizontal surface.
The particle is then acted on by a horizontal force for 3 seconds. This force acts in the direction of motion of the particle and at time $t$ seconds has magnitude $( 3 t + 1 ) \mathrm { N }$. When $t = 0$, the velocity of the particle is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the magnitude of the impulse of the force on the particle between the times $t = 0$ and $t = 3$.
Hence find the velocity of the particle when $t = 3$.
Find the value of $t$ when the velocity of the particle is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

AQA M3 2014 June Q4

4 Two boats, $A$ and $B$, are moving on straight courses with constant speeds. At noon, $A$ and $B$ have position vectors $( \mathbf { i } + 2 \mathbf { j } ) \mathrm { km }$ and $( - \mathbf { i } + \mathbf { j } ) \mathrm { km }$ respectively relative to a lighthouse. Thirty minutes later, the position vectors of $A$ and $B$ are ( $- \mathbf { i } + 3 \mathbf { j }$ ) km and $( 2 \mathbf { i } - \mathbf { j } ) \mathrm { km }$ respectively relative to the lighthouse.

Find the velocity of $A$ relative to $B$ in the form $( m \mathbf { i } + n \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$, where $m$ and $n$ are integers.
The position vector of $A$ relative to $B$ at time $t$ hours after noon is $\mathbf { r } \mathrm { km }$. Show that $$\mathbf { r } = ( 2 - 10 t ) \mathbf { i } + ( 1 + 6 t ) \mathbf { j }$$
Determine the value of $t$ when $A$ and $B$ are closest together.
Find the shortest distance between $A$ and $B$.

AQA M3 2014 June Q5

5 A small smooth ball is dropped from a height of $h$ above a point $A$ on a fixed smooth plane inclined at an angle $\theta$ to the horizontal. The ball falls vertically and collides with the plane at the point $A$. The ball rebounds and strikes the plane again at a point $B$, as shown in the diagram. The points $A$ and $B$ lie on a line of greatest slope of the inclined plane.
\includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-12_318_636_548_712}

Explain whether or not the component of the velocity of the ball parallel to the plane is changed by the collision.
The coefficient of restitution between the ball and the plane is $e$. Find, in terms of $h , \theta , e$ and $g$, the components of the velocity of the ball parallel to and perpendicular to the plane immediately after the collision.
Show that the distance $A B$ is given by $$4 h e ( e + 1 ) \sin \theta$$

AQA M3 2014 June Q6

6 Two smooth spheres, $A$ and $B$, have equal radii and masses 2 kg and 4 kg respectively. The spheres are moving on a smooth horizontal surface and collide. As they collide, $A$ has velocity $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ to the line of centres of the spheres, and $B$ has velocity $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ to the line of centres, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-16_291_844_607_468} Just after the collision, $B$ moves in a direction perpendicular to the line of centres.

Find the speed of $A$ immediately after the collision.
Find the acute angle, correct to the nearest degree, between the velocity of $A$ and the line of centres immediately after the collision.
Find the coefficient of restitution between the spheres.
Find the magnitude of the impulse exerted on $B$ during the collision.

AQA M3 2014 June Q7

4 marks

7 Two small smooth spheres, $A$ and $B$, are the same size and have masses $2 m$ and $m$ respectively. Initially, the spheres are at rest on a smooth horizontal surface. The sphere $A$ receives an impulse of magnitude $J$ and moves with speed $2 u$ directly towards $B$.

$\quad$ Find $J$ in terms of $m$ and $u$.
The sphere $A$ collides directly with $B$. The coefficient of restitution between $A$ and $B$ is $\frac { 2 } { 3 }$. Find, in terms of $u$, the speeds of $A$ and $B$ immediately after the collision.
At the instant of collision, the centre of $B$ is at a distance $s$ from a fixed smooth vertical wall which is at right angles to the direction of motion of $A$ and $B$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_280_1114_1048_497} Subsequently, $B$ collides with the wall. The radius of each sphere is $r$.
Show that the distance of the centre of $A$ from the wall at the instant that $B$ hits the wall is $\frac { 3 s + 12 r } { 5 }$.
The diagram below shows the positions of $A$ and $B$ when $B$ hits the wall.
\includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_330_1109_1822_493} The sphere $B$ collides with $A$ again after rebounding from the wall. The coefficient of restitution between $B$ and the wall is $\frac { 2 } { 5 }$. Find the distance of the centre of $\boldsymbol { B }$ from the wall at the instant when $A$ and $B$ collide again.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-24_2488_1728_219_141}

AQA M3 2015 June Q1

6 marks

1 A formula for calculating the lift force acting on the wings of an aircraft moving through the air is of the form $$F = k v ^ { \alpha } A ^ { \beta } \rho ^ { \gamma }$$ where $F$ is the lift force in newtons,
$k$ is a dimensionless constant,
$v$ is the air velocity in $\mathrm { m } \mathrm { s } ^ { - 1 }$,
$A$ is the surface area of the aircraft's wings in $\mathrm { m } ^ { 2 }$, and
$\rho$ is the density of the air in $\mathrm { kg } \mathrm { m } ^ { - 3 }$.
By using dimensional analysis, find the values of the constants $\alpha , \beta$ and $\gamma$.
[0pt] [6 marks]

AQA M3 2015 June Q2

5 marks

2 A projectile is launched from a point $O$ on top of a cliff with initial velocity $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\alpha$ and moves in a vertical plane. During the motion, the position vector of the projectile relative to the point $O$ is $( x \mathbf { i } + y \mathbf { j } )$ metres where $\mathbf { i }$ and $\mathbf { j }$ are horizontal and vertical unit vectors respectively.

Show that, during the motion, the equation of the trajectory of the projectile is given by $$y = x \tan \alpha - \frac { 4.9 x ^ { 2 } } { u ^ { 2 } \cos ^ { 2 } \alpha }$$
When $u = 21$ and $\alpha = 55 ^ { \circ }$, the projectile hits a small buoy $B$. The buoy is at a distance $s$ metres vertically below $O$ and at a distance $s$ metres horizontally from $O$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-04_601_935_964_548}
1. Find the value of $s$.
2. Find the acute angle between the velocity of the projectile and the horizontal just before the projectile hits $B$, giving your answer to the nearest degree.
  [0pt] [5 marks]

AQA M3 2015 June Q3

4 marks

3 A disc of mass 0.5 kg is moving with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a smooth horizontal surface when it receives a horizontal impulse in a direction perpendicular to its direction of motion. Immediately after the impulse, the disc has speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the magnitude of the impulse received by the disc.
Before the impulse, the disc is moving parallel to a smooth vertical wall, as shown in the diagram. \section*{11/1/1/1/1/1/1/1/1/1/1/1/ Wall} $$\overbrace { 3 \mathrm {~ms} ^ { - 1 } } ^ { \underset { < } { \bigcirc } } \text { Disc }$$ After the impulse, the disc hits the wall and rebounds with speed $3 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the coefficient of restitution between the disc and the wall.
[0pt] [4 marks]

AQA M3 2015 June Q4

2 marks

4 Three uniform smooth spheres, $A , B$ and $C$, have equal radii and masses $m , 2 m$ and $6 m$ 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 $u$ directly towards $B$ and collides with it.
\includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-10_218_1164_500_438} The coefficient of restitution between $A$ and $B$ is $\frac { 2 } { 3 }$.

1. Show that the speed of $B$ immediately after the collision is $\frac { 5 } { 9 } u$.
2. Find, in terms of $u$, the speed of $A$ immediately after the collision.
Subsequently, $B$ collides with $C$. The coefficient of restitution between $B$ and $C$ is $e$. Show that $B$ will collide with $A$ again if $e > k$, where $k$ is a constant to be determined.
Explain why it is not necessary to model the spheres as particles in this question.
[0pt] [2 marks]

AQA M3 2015 June Q5

11 marks

5 Two smooth spheres, $A$ and $B$, have equal radii and masses 2 kg and 1 kg respectively. The spheres move on a smooth horizontal surface and collide. As they collide, $A$ has velocity $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction inclined at an angle $\alpha$ to the line of centres of the spheres, and $B$ has velocity $2.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction inclined at an angle $\beta$ to the line of centres, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-14_458_1068_541_625} The coefficient of restitution between $A$ and $B$ is $\frac { 4 } { 7 }$.
Given that $\sin \alpha = \frac { 4 } { 5 }$ and $\sin \beta = \frac { 12 } { 13 }$, find the speeds of $A$ and $B$ immediately after the collision.
[0pt] [11 marks]

AQA M3 2015 June Q6

18 marks

6 A ship and a navy frigate are a distance of 8 km apart, with the frigate on a bearing of $120 ^ { \circ }$ from the ship, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-16_451_549_411_760} The ship travels due east at a constant speed of $50 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. The frigate travels at a constant speed of $35 \mathrm {~km} \mathrm {~h} ^ { - 1 }$.

1. Find the bearings, to the nearest degree, of the two possible directions in which the frigate can travel to intercept the ship.
  [0pt] [5 marks]
2. Hence find the shorter of the two possible times for the frigate to intercept the ship.
  [0pt] [5 marks]
The captain of the frigate would like the frigate to travel at less than $35 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Find the minimum speed at which the frigate can travel to intercept the ship.
[0pt] [3 marks] $7 \quad$ A particle is projected from a point $O$ on a plane which is inclined at an angle $\theta$ to the horizontal. The particle is projected up the plane with velocity $u$ at an angle $\alpha$ above the horizontal. The particle strikes the plane for the first time at a point $A$. The motion of the particle is in a vertical plane which contains the line $O A$.
\includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-20_469_624_502_685}
Find, in terms of $u , \theta , \alpha$ and $g$, the time taken by the particle to travel from $O$ to $A$.
The particle is moving horizontally when it strikes the plane at $A$. By using the identity $\sin ( P - Q ) = \sin P \cos Q - \cos P \sin Q$, or otherwise, show that $$\tan \alpha = k \tan \theta$$ where $k$ is a constant to be determined.
[0pt] [5 marks]
\includegraphics[max width=\textwidth, alt={}]{bcd20c69-cace-408c-8961-169c19ff0231-24_2488_1728_219_141}

AQA M3 2016 June Q1

2 marks

1 At a firing range, a man holds a gun and fires a bullet horizontally. The bullet is fired with a horizontal velocity of $400 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The mass of the gun is 1.5 kg and the mass of the bullet is 30 grams.

Find the speed of recoil of the gun.
Find the magnitude of the impulse exerted by the man on the gun in bringing the gun to rest after the bullet is fired.
[0pt] [2 marks]

AQA M3 2016 June Q2

3 marks

2 A lunar mapping satellite of mass $m _ { 1 }$ measured in kg is in an elliptic orbit around the moon, which has mass $m _ { 2 }$ measured in kg . The effective potential, $E$, of the satellite is given by $$E = \frac { K ^ { 2 } } { 2 m _ { 1 } r ^ { 2 } } - \frac { G m _ { 1 } m _ { 2 } } { r }$$ where $r$ measured in metres is the distance of the satellite from the moon, $G \mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }$ is the universal gravitational constant, and $K$ is the angular momentum of the satellite. By using dimensional analysis, find the dimensions of:

$E$,
$\quad K$.
[0pt] [3 marks] $3 \quad$ A ball is projected from a point $O$ on horizontal ground with speed $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $30 ^ { \circ }$ above the horizontal. The ball travels in a vertical plane through the point $O$ and hits a point $Q$ on a plane which is inclined at $45 ^ { \circ }$ to the horizontal. The point $O$ is 6 metres from $P$, the foot of the inclined plane, as shown in the diagram. The points $O , P$ and $Q$ lie in the same vertical plane. The line $P Q$ is a line of greatest slope of the inclined plane.
\includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-06_406_1050_568_488}
During its flight, the horizontal and upward vertical distances of the ball from $O$ are $x$ metres and $y$ metres respectively. Show that $x$ and $y$ satisfy the equation $$y = x \frac { \sqrt { 3 } } { 3 } - \frac { x ^ { 2 } } { 30 }$$ Use $\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$ and $\tan 30 ^ { \circ } = \frac { \sqrt { 3 } } { 3 }$.
Find the distance $P Q$.

AQA M3 2016 June Q4

3 marks

4 A smooth uniform sphere $A$, of mass $m$, is moving with velocity $8 u$ in a straight line on a smooth horizontal table. A smooth uniform sphere $B$, of mass $4 m$, has the same radius as $A$ and is moving on the table with velocity $u$.
\includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-10_200_1148_456_447} The sphere $A$ collides directly with the sphere $B$.
The coefficient of restitution between $A$ and $B$ is $e$.

1. Find, in terms of $u$ and $e$, the velocities of $A$ and $B$ immediately after the collision.
2. The direction of motion of $A$ is reversed by the collision. Show that $e > a$, where $a$ is a constant to be determined.
Subsequently, $B$ collides with a fixed smooth vertical wall which is at right angles to the direction of motion of $A$ and $B$. The coefficient of restitution between $B$ and the wall is $\frac { 2 } { 3 }$. The sphere $B$ collides with $A$ again after rebounding from the wall.
Show that $e < b$, where $b$ is a constant to be determined.
Given that $e = \frac { 4 } { 7 }$, find, in terms of $m$ and $u$, the magnitude of the impulse exerted on $B$ by the wall.
[0pt] [3 marks]

AQA M3 2016 June Q5

11 marks

5 A ball is projected from a point $O$ above a smooth plane which is inclined at an angle of $20 ^ { \circ }$ to the horizontal. The point $O$ is at a perpendicular distance of 1 m from the inclined plane. The ball is projected with velocity $22 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $70 ^ { \circ }$ above the horizontal. 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 for the first time at a point $A$.
\includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-14_478_913_571_561}

1. Find the time taken by the ball to travel from $O$ to $A$.
2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at $A$.
After striking $A$, the ball rebounds and strikes the plane for a second time at a point further up than $A$. The coefficient of restitution between the ball and the inclined plane is $e$.
Show that $e < k$, where $k$ is a constant to be determined.
[0pt] [4 marks] $6 \quad$ In this question use $\cos 30 ^ { \circ } = \sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$.
A smooth spherical ball, $A$, is moving with speed $u$ in a straight line on a smooth horizontal table when it hits an identical ball, $B$, which is at rest on the table. Just before the collision, the direction of motion of $A$ is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of $A$ and $B$ makes an angle of $60 ^ { \circ }$ with the wall, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-18_499_1036_721_593} The coefficient of restitution between $A$ and $B$ is $e$.
Show that the speed of $B$ immediately after the collision is $\frac { 1 } { 4 } u ( 1 + e )$ and 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.
Subsequently, $B$ collides with the wall. After colliding with the wall, the direction of motion of $B$ is parallel to the direction of motion of $A$ after its collision with $B$. Show that the coefficient of restitution between $B$ and the wall is $\frac { 1 + e } { 7 - e }$.
[0pt] [7 marks]

AQA M3 2016 June Q7

5 marks

7 A quad-bike, a truck and a car are moving on a large, open, horizontal surface in a desert plain. Relative to the quad-bike, which is travelling due west at its maximum speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the truck is moving on a bearing of $340 ^ { \circ }$. Relative to the car, which is travelling due east at a speed of $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the truck is moving on a bearing of $300 ^ { \circ }$.

Show that the speed of the truck is approximately $24.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and that it is moving on a bearing of $318 ^ { \circ }$, correct to the nearest degree.
At the instant when the truck is at a distance of 400 metres from the quad-bike, the bearing of the truck from the quad-bike is $060 ^ { \circ }$. The truck continues to move with the same velocity as in part (a). The quad-bike continues to move at a speed of $10 \mathrm {~ms} ^ { - 1 }$. Find the bearing, to the nearest degree, on which the quad-bike should travel in order to approach the truck as closely as possible.
[0pt] [5 marks]
\includegraphics[max width=\textwidth, alt={}]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-24_2032_1707_219_153}

AQA M3 2009 June Q1

1 A ball of mass $m$ is travelling vertically downwards with speed $u$ when it hits a horizontal floor. The ball bounces vertically upwards to a height $h$. It is thought that $h$ depends on $m , u$, the acceleration due to gravity $g$, and a dimensionless constant $k$, such that $$h = k m ^ { \alpha } u ^ { \beta } g ^ { \gamma }$$ where $\alpha , \beta$ and $\gamma$ are constants.
By using dimensional analysis, find the values of $\alpha , \beta$ and $\gamma$.

AQA M3 2009 June Q2

2 A particle is projected from a point $O$ on a horizontal plane and has initial velocity components of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $10 \mathrm {~ms} ^ { - 1 }$ parallel to and perpendicular to the plane respectively. At time $t$ seconds after projection, the horizontal and upward vertical distances of the particle from the point $O$ are $x$ metres and $y$ metres respectively.

Show that $x$ and $y$ satisfy the equation $$y = - \frac { g } { 8 } x ^ { 2 } + 5 x$$
By using the equation in part (a), find the horizontal distance travelled by the particle whilst it is more than 1 metre above the plane.
Hence find the time for which the particle is more than 1 metre above the plane.

AQA M3 2009 June Q3

3 A fishing boat is travelling between two ports, $A$ and $B$, on the shore of a lake. The bearing of $B$ from $A$ is $130 ^ { \circ }$. The fishing boat leaves $A$ and travels directly towards $B$ with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A patrol boat on the lake is travelling with speed $4 \mathrm {~ms} ^ { - 1 }$ on a bearing of $040 ^ { \circ }$.
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-3_713_1319_443_406}

Find the velocity of the fishing boat relative to the patrol boat, giving your answer as a speed together with a bearing.
When the patrol boat is 1500 m due west of the fishing boat, it changes direction in order to intercept the fishing boat in the shortest possible time.
1. Find the bearing on which the patrol boat should travel in order to intercept the fishing boat.
2. Given that the patrol boat intercepts the fishing boat before it reaches $B$, find the time, in seconds, that it takes the patrol boat to intercept the fishing boat after changing direction.
3. State a modelling assumption necessary for answering this question, other than the boats being particles.

AQA M3 2009 June Q4

4 A particle of mass 0.5 kg is initially at rest. The particle then moves in a straight line under the action of a single force. This force acts in a constant direction and has magnitude $\left( t ^ { 3 } + t \right) \mathrm { N }$, where $t$ is the time, in seconds, for which the force has been acting.

Find the magnitude of the impulse exerted by the force on the particle between the times $t = 0$ and $t = 4$.
Hence find the speed of the particle when $t = 4$.
Find the time taken for the particle to reach a speed of $12 \mathrm {~ms} ^ { - 1 }$.

AQA M3 2009 June Q5

5 Two smooth spheres, $A$ and $B$, of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, $A$ is moving with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ to the line of centres of the spheres, and $B$ is moving with speed $3 \mathrm {~ms} ^ { - 1 }$ perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_314_1100_593_392} \captionsetup{labelformat=empty} \caption{Before collision}

\end{figure} Immediately after the collision, $A$ and $B$ move with speeds $u$ and $v$ in directions which make angles of $90 ^ { \circ }$ and $40 ^ { \circ }$ respectively with the line of centres, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}

Show that $v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to three significant figures.
Find the coefficient of restitution between the spheres.
Given that the mass of $A$ is 0.5 kg , show that the magnitude of the impulse exerted on $A$ during the collision is 2.17 Ns , correct to three significant figures.
Find the mass of $B$.

AQA M3 2009 June Q6

6 A smooth sphere $A$ of mass $m$ is moving with speed $5 u$ in a straight line on a smooth horizontal table. The sphere $A$ collides directly with a smooth sphere $B$ of mass $7 m$, having the same radius as $A$ and moving with speed $u$ in the same direction as $A$. The coefficient of restitution between $A$ and $B$ is $e$.
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-5_287_880_529_571}

Show that the speed of $B$ after the collision is $\frac { u } { 2 } ( e + 3 )$.
Given that the direction of motion of $A$ is reversed by the collision, show that $e > \frac { 3 } { 7 }$.
Subsequently, $B$ hits a wall fixed at right angles to the direction of motion of $A$ and $B$. The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 2 }$. Given that after $B$ rebounds from the wall both spheres move in the same direction and collide again, show also that $e < \frac { 9 } { 13 }$.
(4 marks)

AQA M3 2009 June Q7

7 A particle is projected from a point $O$ on a smooth plane which is inclined at $30 ^ { \circ }$ to the horizontal. The particle is projected down the plane with velocity $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $40 ^ { \circ }$ above the plane and first strikes it at a point $A$. 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]{719b82f7-2ab5-48db-9b2a-98284096a78a-6_449_963_488_529}

Show that the time taken by the particle to travel from $O$ to $A$ is $$\frac { 20 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
Find the components of the velocity of the particle parallel to and perpendicular to the slope as it hits the slope at $A$.
The coefficient of restitution between the slope and the particle is 0.5 . Find the speed of the particle as it rebounds from the slope.

Questions — AQA M3 (75 questions)

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AQA M3 2009 June Q2

AQA M3 2009 June Q3

AQA M3 2009 June Q4

AQA M3 2009 June Q5

AQA M3 2009 June Q6

AQA M3 2009 June Q7