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AQA Further AS Paper 2 Mechanics 2022 June Q20

1 marks Easy -1.8

20 J
25 J
50 J
100 J 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] $\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ $\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ $\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }$ $\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }$ 3 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of $12 \mathrm {~ms} ^ { - 1 }$ The ball is thrown from ground level. 3

Calculate the initial kinetic energy of the ball. 3
The maximum height of the ball above the ground is $h$ metres.
Jeff and Gurjas use an energy method to find $h$ Jeff concludes that $h = 7.3$ Gurjas concludes that $h < 7.3$ Explain the reasoning that they have used, showing any calculations that you make.
4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
Two of these classifications are deep water waves and shallow water waves.
4
The wavelength $w$ of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where $g$ is the acceleration due to gravity and $t$ is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that $k$ is a dimensionless constant. 4
The wavelength $w$ of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where $g$ is the acceleration due to gravity, $d$ is the depth of water and $t$ is the time period between consecutive waves. Use dimensional analysis to find the values of $\alpha$ and $\beta$ 5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of $v \mathrm {~ms} ^ { - 1 }$, it experiences a resistance force of magnitude $25 v$ newtons. The car has a maximum speed of $72 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ on the straight road.
Find the maximum power output of the car.
Fully justify your answer.

Find the value of $\omega$ 4
1. Find the magnitude of the horizontal resultant force acting on Reena.
  4
(ii) Show the direction of this horizontal resultant force on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-03_380_442_2017_861} 5 An impulse of $\left[ \begin{array} { r } - 5 \\ 12 \end{array} \right] \mathrm { N } \mathrm { s }$ is applied to a particle of mass 5 kg which is moving with velocity $\left[ \begin{array} { l } 6 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ 5
Calculate the magnitude of the impulse. 5
Find the speed of the particle immediately after the impulse is applied.

AQA Further AS Paper 2 Mechanics 2023 June Q6

4 marks Moderate -0.5

6 A ball is thrown with speed $u$ at an angle of $45 ^ { \circ }$ to the horizontal from a point $O$ When the horizontal displacement of the ball is $x$, the vertical displacement of the ball above $O$ is $y$ where $$y = x - \frac { k x ^ { 2 } } { u ^ { 2 } }$$ 6

Use dimensional analysis to find the dimensions of $k$ 6
State what can be deduced about $k$ from the dimensions that you found in part (a).

AQA Further AS Paper 2 Mechanics 2023 June Q7

6 marks Standard +0.3

7 Two smooth, equally sized spheres, $A$ and $B$, are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
Immediately after the collision, $A$ has speed $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $B$ has speed $3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ The coefficient of restitution between the spheres is $e$ 7

1. Show that $A$ does not change its direction of motion as a result of the collision.
  7
(ii) Find the value of $e$ 7
Given that the mass of $B$ is 0.6 kg , find the mass of $A$

AQA Further AS Paper 2 Mechanics 2023 June Q8

7 marks Challenging +1.2

8 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ Omar, a bungee jumper of mass 70 kg , has his ankles attached to one end of an elastic cord. The other end of the cord is attached to a bridge which is 80 metres above the surface of a river. Omar steps off the bridge at the point where the cord is attached and falls vertically downwards. The cord can be modelled as a light elastic string of natural length $L$ metres and modulus of elasticity 2800 N Model Omar as a particle. 8

Given that Omar just reaches the surface of the river before being pulled back up, find the value of $L$ Fully justify your answer.
8
If Omar is not modelled as a particle, explain the effect of revising this assumption on your answer to part (a).

AQA Further AS Paper 2 Mechanics 2023 June Q18

1 marks Easy -1.8

18 J
34 J 2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer. $e = - 1$ $e = 0$ $e = \frac { 1 } { 2 }$ $e = 1$ 3 A stone of mass 0.2 kg is thrown vertically upwards with a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find the initial kinetic energy of the stone.
Circle your answer.
[0pt] [1 mark]
1 J
5 J
10 J

AQA Further AS Paper 2 Mechanics 2023 June Q20

Easy -1.2

20 J 4 Reena is skating on an ice rink, which has a horizontal surface. She follows a circular path of radius 5 metres and centre $O$ She completes 10 full revolutions in 1 minute, moving with a constant angular speed of $\omega$ radians per second. The mass of Reena is 40 kg
4

Find the value of $\omega$ 4
1. Find the magnitude of the horizontal resultant force acting on Reena.
  4
(ii) Show the direction of this horizontal resultant force on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-03_380_442_2017_861} 5 An impulse of $\left[ \begin{array} { r } - 5 \\ 12 \end{array} \right] \mathrm { N } \mathrm { s }$ is applied to a particle of mass 5 kg which is moving with velocity $\left[ \begin{array} { l } 6 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ 5
Calculate the magnitude of the impulse. 5
Find the speed of the particle immediately after the impulse is applied.
6 A ball is thrown with speed $u$ at an angle of $45 ^ { \circ }$ to the horizontal from a point $O$ When the horizontal displacement of the ball is $x$, the vertical displacement of the ball above $O$ is $y$ where $$y = x - \frac { k x ^ { 2 } } { u ^ { 2 } }$$ 6
Use dimensional analysis to find the dimensions of $k$ 6
State what can be deduced about $k$ from the dimensions that you found in part (a).
7 Two smooth, equally sized spheres, $A$ and $B$, are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
Immediately after the collision, $A$ has speed $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $B$ has speed $3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ The coefficient of restitution between the spheres is $e$ 7
1. Show that $A$ does not change its direction of motion as a result of the collision.
  7
(ii) Find the value of $e$ 7
Given that the mass of $B$ is 0.6 kg , find the mass of $A$

AQA Further AS Paper 2 Mechanics 2024 June Q1

1 marks Easy -1.2

1 An elastic string has modulus of elasticity 20 newtons and natural length 2 metres.
The string is stretched so that its extension is 0.5 metres.
Find the elastic potential energy stored in the string.
Circle your answer.
[0pt] [1 mark]
1.25 J
5.5 J
5 J
10 J

AQA Further AS Paper 2 Mechanics 2024 June Q2

1 marks Easy -2.0

2 State the dimensions of impulse.
Circle your answer.
[0pt] [1 mark] $M L T ^ { - 2 }$ $M L T ^ { - 1 }$ MLT $M L T { } ^ { 2 }$

AQA Further AS Paper 2 Mechanics 2024 June Q3

1 marks Easy -1.2

3 A cyclist travels around a circular track of radius 20 m at a constant speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find the angular speed of the cyclist in radians per second. Circle your answer.
[0pt] [1 mark] $0.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $0.4 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $3.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$

AQA Further AS Paper 2 Mechanics 2024 June Q5

4 marks Easy -1.8

5 J
10 J 2 State the dimensions of impulse.
Circle your answer.
[0pt] [1 mark] $M L T ^ { - 2 }$ $M L T ^ { - 1 }$ MLT $M L T { } ^ { 2 }$ 3 A cyclist travels around a circular track of radius 20 m at a constant speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find the angular speed of the cyclist in radians per second. Circle your answer.
[0pt] [1 mark] $0.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $0.4 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $3.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$

AQA Further AS Paper 2 Mechanics 2024 June Q6

4 marks Moderate -0.8

6 Kepler's Third Law of planetary motion for the period of a circular orbit around the Earth is given by the formula, $$t = 2 \pi \sqrt { \frac { r ^ { 3 } } { G m } }$$ where, $t$ is the time taken for one orbit $r$ is the radius of the circular orbit $m$ is the mass of the Earth $G$ is a gravitational constant. Use dimensional analysis to determine the dimensions of $G$ \includegraphics[max width=\textwidth, alt={}, center]{ce05dedd-515b-49e2-92f3-f5ec22bab4be-08_2491_1755_173_123}

AQA Further AS Paper 2 Mechanics 2024 June Q7

5 marks Standard +0.3

7 A single force, $F$ newtons, acts on a particle moving on a straight, smooth, horizontal line. The force $F$ acts in the direction of motion of the particle.
At time $t$ seconds, $F = 6 \mathrm { e } ^ { t } + 2 \mathrm { e } ^ { 2 t }$ where $0 \leq t \leq \ln 8$ 7

Find the impulse of $F$ over the interval $0 \leq t \leq \ln 8$ 7
The particle has a mass of 2 kg and at time $t = 0$ has velocity $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find the velocity of the particle when $t = \ln 8$

AQA Further AS Paper 2 Mechanics 2024 June Q8

10 marks Standard +0.3

8 Two spheres, $A$ and $B$, of equal size are moving in the same direction along a straight line on a smooth horizontal surface. Sphere $A$ has mass $m$ and is moving with speed $4 u$ Sphere $B$ has mass $6 m$ and is moving with speed $u$ The diagram shows the spheres and their velocities. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ce05dedd-515b-49e2-92f3-f5ec22bab4be-10_227_446_648_781} \captionsetup{labelformat=empty} \caption{A}

\end{figure} B Subsequently $A$ collides directly with $B$ The coefficient of restitution between $A$ and $B$ is $e$ 8

Find, in terms of $m$ and $u$, the total momentum of the spheres before the collision.
8
Show that the speed of $B$ immediately after the collision is $\frac { u ( 3 e + 10 ) } { 7 }$ 8
After the collision sphere $A$ moves in the opposite direction.
Find the range of possible values for $e$ [0pt] [5 marks]

AQA Further AS Paper 2 Mechanics 2024 June Q10

2 marks Easy -1.8

10 J 2 State the dimensions of impulse.
Circle your answer.
[0pt] [1 mark] $M L T ^ { - 2 }$ $M L T ^ { - 1 }$ MLT $M L T { } ^ { 2 }$ 3 A cyclist travels around a circular track of radius 20 m at a constant speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find the angular speed of the cyclist in radians per second. Circle your answer.
[0pt] [1 mark] $0.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $0.4 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }$ $3.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$

AQA Further AS Paper 2 Discrete 2018 June Q1

1 marks Easy -1.2

1 The table shows some of the outcomes of performing a modular arithmetic operation.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	2	3
2		1
3	1

Which pair are operations that could each be represented by the table?
Tick ( ✓ ) one box. \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_109_111_1338_497} Addition $\bmod 6$ and multiplication $\bmod 5$ \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_108_109_1471_497} Addition mod 6 and multiplication $\bmod 6$ \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_113_109_1603_497} Addition mod 4 and multiplication $\bmod 5$ \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_107_109_1742_497} Addition mod 4 and multiplication mod 6

AQA Further AS Paper 2 Discrete 2018 June Q2

1 marks Standard +0.3

2 The binary operation ⊗ is given by $a \otimes b = 3 a ( 5 + b ) ( \bmod 8 )$ where $a , b \in \mathbb { Z }$ Given that $2 \otimes x = 6$, which of the integers below is a possible value of $x$ ?
Circle your answer.
[0pt] [1 mark]
0123

AQA Further AS Paper 2 Discrete 2018 June Q3

4 marks Moderate -0.5

3 Alex and Sam are playing a zero-sum game. The game is represented by the pay-off matrix for Alex.

Sam
\cline { 2 - 5 }	Strategy
\cline { 2 - 5 }	$\mathbf { S } _ { \mathbf { 1 } }$	$\mathbf { S } _ { \mathbf { 2 } }$	$\mathbf { S } _ { \mathbf { 3 } }$
$\mathbf { A } _ { \mathbf { 1 } }$	2	2	3
\cline { 2 - 5 }	$\mathbf { A } _ { \mathbf { 2 } }$	0	3	5
$\mathbf { A } _ { \mathbf { 3 } }$	- 1	2	- 2

Explain why the value of the game is 2
3
Identify the play-safe strategy for each player.
Each pipe is labelled with its upper capacity in $\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }$ \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-04_620_940_450_550}

AQA Further AS Paper 2 Discrete 2018 June Q4

6 marks Moderate -0.5

1. Find the value of the cut given by $\{ A , B , C , D , F , J \} \{ E , G , H \}$.
  4
(ii) State what can be deduced about the maximum flow through the network.
4
1. List the nodes which are sources of the network. 4
(ii) Add a supersource $S$ to the network. 4
1. List the nodes which are sinks of the network. 4
(ii) Add a supersink $T$ to the network.

AQA Further AS Paper 2 Discrete 2018 June Q5

9 marks Moderate -0.5

5 A group of friends want to prepare a meal. They start preparing the meal at 6:30 pm Activities to prepare the meal are shown in Figure 1 below. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 1}

Label	Activity	Duration (mins)	Immediate predecessors
A	Weigh rice	1	-
$B$	Cook rice	18	$A$
C	Drain rice	1	B
D	Chop vegetables	10	-
$E$	Fry vegetables	12
$F$	Combine fried vegetables and drained rice	1
G	Prepare sauce ingredients	4	-
$H$	Boil sauce	12
$I$	Serve meal on plates	2

\end{table} 5

1. Use Figure 2 shown below to complete Figure 1 above. 5
(ii) Complete Figure 2 showing the earliest start time and latest finishing time for each activity. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-06_700_1650_1781_194}
\end{figure} 5
1. State the activity which must be started first so that the meal is served in the shortest possible time. Fully justify your answer.
  5
(ii) Determine the earliest possible time at which the preparation of the meal can be completed.
Question 5 continues on the next page 5
The group of friends want to cook spring rolls so that they are served at the same time as the rest of the meal. This requires the additional activities shown in Figure 3. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 3}
Label Activity Duration Immediate predecessors
J Switch on and heat oven -
K Put spring rolls in oven and cook
$L$ Transfer spring rolls to serving dish
\end{table} It takes 15 seconds to switch on the oven. The oven must be allowed to heat up for 10 minutes before the spring rolls are put in the oven. It takes 15 seconds to put the spring rolls in the oven.
The spring rolls must cook in the hot oven for 8 minutes.
It takes 30 seconds to transfer the spring rolls to a serving dish.
5
1. Complete Figure 3 above. 5
(ii) Determine the latest time at which the oven can be switched on in order for the spring rolls to be served at the same time as the rest of the meal.
[0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-09_2488_1716_219_153}

AQA Further AS Paper 2 Discrete 2018 June Q6

5 marks Moderate -0.8

6 An animal sanctuary has a rainwater collection site. The manager of the sanctuary is installing a pipe system to connect the rainwater collection site to five other sites in the sanctuary. Each site does not need to be connected directly to the rainwater collection site. There are nine possible routes between the sites that are suitable for water pipes. The distances, in metres, of the nine possible routes are given in the table below.

From/To	Henhouse (H)	Goatshed (G)	Kennels (K)	Cattery (C)
Rainwater collection site (R)	840	810	520	370
Cattery (C)	-	680	610	\multirow{3}{*}{}
Duckpond (D)	480	310
Goatshed (G)	150

Water pipe costs 60 pence per metre. Find the minimum cost of connecting all the sites to the rainwater collection site. Fully justify your answer. $7 \quad$ A linear programming problem has the constraints $$\begin{aligned} 1 \leq x & \leq 6 \\ 1 \leq y & \leq 6 \\ y & \geq x \\ x + y & \leq 11 \end{aligned}$$

AQA Further AS Paper 2 Discrete 2018 June Q7

14 marks Standard +0.8

1. Complete Figure 4 to identify the feasible region for the problem. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-12_922_940_849_552}
  \end{figure} 7
(ii) Determine the maximum value of $5 x + 4 y$ subject to the constraints.
7
The simple-connected graph $G$ has seven vertices. The vertices of $G$ have degree $1,2,3 , v , w , x$ and $y$ 7
1. Explain why $x \geq 1$ and $y \geq 1$ 7
(ii) Explain why $x \leq 6$ and $y \leq 6$ 7
(iii) Explain why $x + y \leq 11$ 7
(iv) State an additional constraint that applies to the values of $x$ and $y$ in this context.
7
The graph $G$ also has eight edges. The inequalities used in part (a)(i) apply to the graph $G$. 7
1. Given that $v + w = 4$, find all the feasible values of $x$ and $y$.
  7
(ii) It is also given that the graph $G$ is semi-Eulerian. On Figure 5, draw $G$. Figure 5

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Label	Activity	Duration	Immediate predecessors
J	Switch on and heat oven		-
K	Put spring rolls in oven and cook
\(L\)	Transfer spring rolls to serving dish

Sam
\cline { 2 - 5 }	Strategy
\cline { 2 - 5 }	\(\mathbf { S } _ { \mathbf { 1 } }\)	\(\mathbf { S } _ { \mathbf { 2 } }\)	\(\mathbf { S } _ { \mathbf { 3 } }\)
\(\mathbf { A } _ { \mathbf { 1 } }\)	2	2	3
\cline { 2 - 5 }	\(\mathbf { A } _ { \mathbf { 2 } }\)	0	3	5
\(\mathbf { A } _ { \mathbf { 3 } }\)	- 1	2	- 2

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