Questions Further AS Paper 2 Mechanics

AQA Further AS Paper 2 Mechanics 2020 June Q1

1 marks Moderate -0.8

1 In this question use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A particle of mass 2 kg is attached to one end of a light elastic string of natural length 0.5 metres and modulus of elasticity 100 N . The other end of the string is attached to the point $O$. Find the extension of the elastic string when the particle hangs in equilibrium vertically below $O$. Circle your answer.
0.01 m
0.1 m
0.2 m
0.4 m

AQA Further AS Paper 2 Mechanics 2020 June Q2

1 marks Moderate -0.8

2 An object moves under the action of a single force $F$ newtons.
It is given that $F = 6 x ^ { 2 }$, where $x$ represents the displacement in metres from the initial position of the object. Find the work done by $F$ in moving the object from $x = 1$ to $x = 2$ Circle your answer.
[0pt] [1 mark]
12 J
14 J
18J
42 J

AQA Further AS Paper 2 Mechanics 2020 June Q3

3 marks Easy -1.2

3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of $3.84 \times 10 ^ { 8 }$ metres.
Find the approximate speed of the moon relative to Earth, in metres per second.

AQA Further AS Paper 2 Mechanics 2020 June Q4

4 marks Moderate -0.3

4 A particle $P$, of mass $m \mathrm {~kg}$, collides with a particle $Q$, of mass 2 kg Immediately before the collision the velocity of $P$ is $\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ As a result of the collision the particles coalesce into a single particle which moves with velocity $\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant. Find the value of $k$.

AQA Further AS Paper 2 Mechanics 2020 June Q5

5 marks Moderate -0.3

5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is $120 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ Find the maximum power output of the engine.
Fully justify your answer.

AQA Further AS Paper 2 Mechanics 2020 June Q6

5 marks Standard +0.3

6 The magnitude of the gravitational force $F$ between two planets of masses $m _ { 1 }$ and $m _ { 2 }$ with centres at a distance $d$ apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where $G$ is a constant.
6

Show that $G$ must have dimensions $L ^ { 3 } M ^ { - 1 } T ^ { - 2 }$, where $L$ represents length, $M$ represents mass and $T$ represents time.
6
The lifetime $t$ of a planet is thought to depend on its mass $m$, its radius $r$, the constant $G$ and a dimensionless constant $k$ such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where $a , b$ and $c$ are constants.
Determine the values of $a , b$ and $c$.

AQA Further AS Paper 2 Mechanics 2020 June Q7

9 marks Standard +0.3

7 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ As part of a competition, Jo-Jo makes a small pop-up rocket.
It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
The mass of the rocket is 18 grams and the stiffness constant of the spring is $60 \mathrm { Nm } ^ { - 1 }$ Initially the spring is compressed by 3 cm
7

Find the speed of the rocket when the spring first reaches its natural length.
7
By considering energy find the distance that the rocket rises. 7
In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.

AQA Further AS Paper 2 Mechanics 2020 June Q14

4 marks Moderate -0.5

14 J
18J
42 J 3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of $3.84 \times 10 ^ { 8 }$ metres.
Find the approximate speed of the moon relative to Earth, in metres per second.
4 A particle $P$, of mass $m \mathrm {~kg}$, collides with a particle $Q$, of mass 2 kg Immediately before the collision the velocity of $P$ is $\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ As a result of the collision the particles coalesce into a single particle which moves with velocity $\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant. Find the value of $k$.
5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is $120 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ Find the maximum power output of the engine.
Fully justify your answer.
6 The magnitude of the gravitational force $F$ between two planets of masses $m _ { 1 }$ and $m _ { 2 }$ with centres at a distance $d$ apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where $G$ is a constant.
6

Show that $G$ must have dimensions $L ^ { 3 } M ^ { - 1 } T ^ { - 2 }$, where $L$ represents length, $M$ represents mass and $T$ represents time.
6

The lifetime $t$ of a planet is thought to depend on its mass $m$, its radius $r$, the constant $G$ and a dimensionless constant $k$ such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where $a , b$ and $c$ are constants.
Determine the values of $a , b$ and $c$.
7 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ As part of a competition, Jo-Jo makes a small pop-up rocket.
It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
The mass of the rocket is 18 grams and the stiffness constant of the spring is $60 \mathrm { Nm } ^ { - 1 }$ Initially the spring is compressed by 3 cm
7

Find the speed of the rocket when the spring first reaches its natural length.
7
By considering energy find the distance that the rocket rises. 7
In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.
8 Two smooth spheres $A$ and $B$ have the same radius and are free to move on a smooth horizontal surface. The masses of $A$ and $B$ are $2 m$ and $m$ respectively.
Both $A$ and $B$ are initially at rest.
The sphere $A$ is set in motion directly towards $B$ with speed $3 u$ and at the same time $B$ is set in motion directly towards $A$ with speed $2 u$. Subsequently $A$ and $B$ collide directly. $A$ The coefficient of restitution between the spheres is $e$.
8
Show that the speed of $B$ after the collision is given by $$\frac { 2 u ( 2 + 5 e ) } { 3 }$$ \section*{Question 8 continues on the next page} 8
Given that the direction of the velocity of $A$ is reversed during the collision, find the range of possible values of $e$. Fully justify your answer.
[0pt] [4 marks]
8

Given that the magnitude of the impulse that $A$ exerts on $B$ is $\frac { 19 m u } { 3 }$, find the value of $e$.

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AQA Further AS Paper 2 Mechanics Specimen Q1

1 marks Easy -1.2

1 A child, of mass 40 kg , moves at constant speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a fairground ride.
The path of the child is a circle of radius 4 metres.
Find the magnitude of the resultant force acting on the child.
Circle your answer.
[0pt] [1 mark]
6.3 N
50 N
130 N
250 N

AQA Further AS Paper 2 Mechanics Specimen Q2

1 marks Easy -1.2

2 The graph shows how a force, $F$, varies with time during a period of 0.8 seconds. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-03_440_960_568_516} Find the magnitude of the impulse of $F$ during the 0.8 seconds.
Circle your answer.
[0pt] [1 mark]
1.0 Ns
1.6 Ns
2.2 Ns
3.2 Ns Turn over for the next question

AQA Further AS Paper 2 Mechanics Specimen Q3

4 marks Moderate -0.3

3 A tank full of liquid has a hole made in its base.
Two students, Sarah and David, propose two different models for the speed, $v$, at which liquid exits the tank. David thinks that $v$ will depend on the height of the liquid in the tank, $h$, the acceleration due to gravity, $g$, and the density of the liquid, $\rho$, such that $v \propto g ^ { a } h ^ { b } \rho ^ { c }$ where $a$, $b$ and $c$ are constants. Sarah thinks that $v$ will not depend on the density of the liquid and suggests the model $v \propto g ^ { a } h ^ { b }$ 3

By considering dimensions, explain which student's model should be rejected.
[0pt] [2 marks]
3
Find the values of the constants in order for the model that you did not reject in part (a) to be dimensionally consistent.
[0pt] [2 marks]

AQA Further AS Paper 2 Mechanics Specimen Q4

5 marks Moderate -0.3

4 A cricket ball of mass 156 grams is thrown from a point which is 1.5 metres above the ground, with a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ A tennis ball of mass 58 grams is thrown from the same point, with the same speed.
Prove that both balls hit the ground with the same speed.
Clearly state any assumptions you have made and how you have used them.
[0pt] [5 marks]

AQA Further AS Paper 2 Mechanics Specimen Q5

4 marks Standard +0.3

5 Two small smooth discs, $C$ and $D$, have equal radii and masses of 2 kg and 3 kg respectively. The discs are sliding on a smooth horizontal surface towards each other and collide directly. Disc $C$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and disc $D$ has speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ as they collide. The coefficient of restitution between $C$ and $D$ is 0.6 The diagram shows the discs, viewed from above, before the collision. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-06_343_712_868_753} 5

Show that the speed of $D$ immediately after the collision is $1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 2 significant figures.
5
Find the speed of $C$ immediately after the collision.
[0pt] [2 marks]
5
In fact the horizontal surface on which the discs are sliding is not smooth.
Explain how the introduction of friction will affect your answer to part (b).
[0pt] [2 marks]
Turn over for the next question

AQA Further AS Paper 2 Mechanics Specimen Q6

4 marks Standard +0.3

6 A car, of mass 1200 kg , moves on a straight horizontal road where it has a maximum speed of $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ When the car travels at a speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ it experiences a resistance force which can be modelled as being of magnitude 30 v newtons. 6

Show that the power output of the car is 48000 W , when it is travelling at its maximum speed. 6
Find the maximum acceleration of the car when it is travelling at a speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ [0pt] [4 marks]

AQA Further AS Paper 2 Mechanics Specimen Q7

3 marks Standard +0.3

7 A disc, of mass 0.15 kg , slides across a smooth horizontal table and collides with a vertical wall which is perpendicular to the path of the disc. The disc is in contact with the wall for 0.02 seconds and then rebounds.
A possible model for the force, $F$ newtons, exerted on the disc by the wall, whilst in contact, is given by $$F = k t ^ { 2 } ( t - b ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.020$$ where $k$ and $b$ are constants.
The force is initially zero and becomes zero again as the disc loses contact with the wall. 7

State the value of $b$.
7
Find the magnitude of the impulse on the disc, giving your answer in terms of $k$.
7
The disc is travelling at $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it hits the wall.
The disc rebounds with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find $k$.
[0pt] [3 marks]

AQA Further AS Paper 2 Mechanics Specimen Q8

6 marks Challenging +1.2

8 In this question use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
A particle, of mass 2 kg , is attached to one end of a light elastic string of natural length 0.2 metres. The other end of the string is attached to a fixed point $O$.
The particle is pulled down and released from rest at a point 0.6 metres directly below $O$.
The particle then moves vertically and next comes to rest when it is 0.1 metres below $O$.
Assume that no air resistance acts on the particle.
8

Find the maximum speed of the particle.
[0pt] [6 marks]
8
Describe one way in which the model you have used could be refined.

AQA Further AS Paper 2 Mechanics 2018 June Q1

1 marks Easy -1.8

1 A particle $A$, of mass 0.2 kg , collides with a particle $B$, of mass 0.3 kg Immediately before the collision, the velocity of $A$ is $\left[ \begin{array} { c } 4 \\ 12 \end{array} \right] \mathrm { ms } ^ { - 1 }$ and the velocity of $B$ is $\left[ \begin{array} { l } - 1 \\ - 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle.
Circle your answer.
[0pt] [1 mark] $$\left[ \begin{array} { l } 0.5 \\ 1.5 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 2 \\ 6 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 } \quad \left[ \begin{array} { l } 3 \\ 9 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 }$$

AQA Further AS Paper 2 Mechanics 2018 June Q2

1 marks Easy -1.2

2 A train is travelling at maximum speed with its engine using its maximum power of 1800 kW When travelling at this speed the train experiences a total resistive force of 40000 N Find the maximum speed of the train. Circle your answer.
[0pt] [1 mark] $22 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ $45 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ $54 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ $90 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

AQA Further AS Paper 2 Mechanics 2018 June Q3

5 marks Standard +0.3

3 The kinetic energy, $E$, of a compound pendulum is given by $$E = \frac { 1 } { 2 } I \omega ^ { 2 }$$ where $\omega$ is the angular speed and $I$ is a quantity called the moment of inertia.
3

Show that for this formula to be dimensionally consistent then $I$ must have dimensions $M L ^ { 2 }$, where $M$ represents mass and $L$ represents length.
[0pt] [2 marks]
3
The time, $T$, taken for one complete swing of a pendulum is thought to depend on its moment of inertia, $I$, its weight, $W$, and the distance, $h$, of the centre of mass of the pendulum from the point of suspension. The formula being proposed is $$T = k I ^ { \alpha } W ^ { \beta } h ^ { \gamma }$$ where $k$ is a dimensionless constant. Determine the values of $\alpha , \beta$ and $\gamma$.

AQA Further AS Paper 2 Mechanics 2018 June Q4

11 marks Standard +0.8

4 Two smooth spheres $A$ and $B$ of equal radius are free to move on a smooth horizontal surface. The masses of $A$ and $B$ are $m$ and $4 m$ respectively.
The coefficient of restitution between the spheres is $e$.
The spheres are projected directly towards each other, each with speed $u$, and subsequently collide. 4

Show that the speed of $B$ immediately after the impact with $A$ is $$\frac { u ( 3 - 2 e ) } { 5 }$$ 4
Find the speed of $A$ in terms of $u$ and $e$.
4
Comment on the direction of motion of the spheres after the collision, justifying your answer.
4
The magnitude of the impulse on $B$ due to the collision is $I$.
Deduce that $$\frac { 8 m u } { 5 } \leq I \leq \frac { 16 m u } { 5 }$$

AQA Further AS Paper 2 Mechanics 2018 June Q5

6 marks Standard +0.3

5 A car travels around a roundabout at a constant speed. The surface of the roundabout is horizontal. The car has mass 990 kg and the path of the car is a circular arc of radius 48 metres.
A simple model assumes that the car is a particle and the only horizontal force acting on it as it travels around the roundabout is friction. On a dry day typical values of friction, $F$, between the surface of the roundabout and the tyres of the car are $$7300 \mathrm {~N} \leq F \leq 9200 \mathrm {~N}$$ 5

Using this model calculate a safe speed limit, in miles per hour, for the car as it travels around the roundabout. Explain your reasoning fully.
Note that there are 1600 metres in one mile.
5
Gary assumes that on a wet day typical values for friction, $F$, are $$5400 \mathrm {~N} \leq F \leq 10000 \mathrm {~N}$$ Comment on the validity of Gary's revised assumption.

AQA Further AS Paper 2 Mechanics 2018 June Q6

7 marks Standard +0.3

6 At a fairground a dodgem car is moving in a straight horizontal line towards a side wall that is perpendicular to the velocity of the car. The speed of the car is $1.8 \mathrm {~ms} ^ { - 1 }$ It collides with the side wall and rebounds along its original path with a speed of $1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ The total mass of the dodgem car and the passengers is 250 kg
6

Find the magnitude of the impulse on the car during the collision with the side wall.
6
A possible model for the magnitude of the force, $F$ newtons, acting on the dodgem car due to its collision with the side wall is given by $$F = k t ( 4 - 5 t ) \quad \text { for } 0 \leq t \leq 0.8$$ 6 (b) (i) Find the value of $k$.
(b) (ii) Determine the maximum magnitude of the force predicted by the model. 6 (b) (ii) Determine the maximum magnitude of the fored bed bed at

AQA Further AS Paper 2 Mechanics 2018 June Q7

9 marks Standard +0.3

7

Find Dominic's speed at the point when the cord initially becomes taut.
7
Determine whether or not Dominic enters the river and gets wet.
7
One limitation of this model is that Dominic is not a particle.
Explain the effect of revising this assumption on your answer to part (b). \includegraphics[max width=\textwidth, alt={}, center]{1b79a789-c003-46c9-9235-254c1d8a0501-12_2492_1721_217_150} Question number Additional page, if required.
Write the question numbers in the left-hand margin. Question number Additional page, if required.
Write the question numbers in the left-hand margin. Additional page, if required.
Write the question numbers in the left-hand margin.

AQA Further AS Paper 2 Mechanics 2022 June Q1

1 marks Easy -1.8

1 A box is being pushed in a straight line along horizontal ground by a force.
The force is applied in the direction of motion and has magnitude 10 newtons. The box moves 5 metres in 2 seconds. Calculate the work done by the force.
Circle your answer.
20 J
25 J
50 J
100 J

AQA Further AS Paper 2 Mechanics 2022 June Q2

1 marks Easy -1.8

2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are $\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }$ and $\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }$ As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark] $\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ $\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ $\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }$ $\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }$

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Question number	Additional page, if required. Write the question numbers in the left-hand margin.
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