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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
  1. 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
  2. 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
  1. Show that the speed of \(B\) immediately after the impact with \(A\) is $$\frac { u ( 3 - 2 e ) } { 5 }$$ 4
  2. Find the speed of \(A\) in terms of \(u\) and \(e\).
    4
  3. Comment on the direction of motion of the spheres after the collision, justifying your answer.
    4
  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
  1. 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
  2. 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
  1. Find the magnitude of the impulse on the car during the collision with the side wall.
    6
  2. 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)
    1. Find the value of \(k\).
    2. 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
  1. Find Dominic's speed at the point when the cord initially becomes taut.
    7
  2. Determine whether or not Dominic enters the river and gets wet.
    7
  3. 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.
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25 J
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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] \(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
AQA Further AS Paper 2 Mechanics 2022 June Q3
4 marks Moderate -0.3
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
  1. Calculate the initial kinetic energy of the ball. 3
  2. 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.
AQA Further AS Paper 2 Mechanics 2022 June Q4
5 marks Moderate -0.8
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
  1. 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
  2. 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 \(\beta\)
AQA Further AS Paper 2 Mechanics 2022 June Q5
5 marks Moderate -0.3
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.
AQA Further AS Paper 2 Mechanics 2022 June Q6
7 marks Standard +0.3
6 An ice hockey puck, of mass 0.2 kg , is moving in a straight line on a horizontal ice rink under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude ( \(2 t + 3\) ) newtons.
The force acts on the puck from \(t = 0\) to \(t = T\) 6
  1. Show that the magnitude of the impulse of the force is \(a T ^ { 2 } + b T\), where \(a\) and \(b\) are integers to be found.
    [0pt] [3 marks]
    6
  2. While the force acts on the puck, its speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Use your answer from part (a) to find \(T\), giving your answer to three significant figures.
    Fully justify your answer.
AQA Further AS Paper 2 Mechanics 2022 June Q7
9 marks Standard +0.3
7 The particles \(A\) and \(B\) are moving on a smooth horizontal surface directly towards each other. Particle \(A\) has mass 0.4 kg and particle \(B\) has mass 0.2 kg
Particle \(A\) has speed \(4 \mathrm {~ms} ^ { - 1 }\) and particle \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) when they collide, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ec39a757-5867-4798-b26c-73cd5746581c-08_392_1064_625_488} The coefficient of restitution between the particles is \(e\) 7
  1. Find the magnitude of the total momentum of the particles before the collision.
    [0pt] [2 marks] 7
  2. (i) Show that the speed of \(B\) immediately after the collision is \(( 4 e + 2 ) \mathrm { ms } ^ { - 1 }\) [0pt] [3 marks]
    7 (b) (ii) Find an expression, in terms of \(e\), for the speed of \(A\) immediately after the collision.
    7
  3. Explain what happens to particle \(A\) when the collision is perfectly elastic.
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
  1. Calculate the initial kinetic energy of the ball. 3
  2. 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
    1. 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
    2. 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.
AQA Further AS Paper 2 Mechanics 2023 June Q1
1 marks Standard +0.3
1 A particle moves along the \(x\)-axis under the action of a force, \(F\) newtons, where $$F = 3 x ^ { 2 } + 5$$ Find the work done by the force as the particle moves from \(x = 0\) metres to \(x = 2\) metres. Circle your answer.
12 J
17 J
18 J
34 J
AQA Further AS Paper 2 Mechanics 2023 June Q2
1 marks Easy -2.0
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. \(e = - 1\) \(e = 0\) \(e = \frac { 1 } { 2 }\) \(e = 1\)
AQA Further AS Paper 2 Mechanics 2023 June Q3
1 marks Easy -1.8
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
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AQA Further AS Paper 2 Mechanics 2023 June Q5
4 marks Moderate -0.8
5 J
10 J
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
  1. Find the value of \(\omega\) 4
  2. (i) Find the magnitude of the horizontal resultant force acting on Reena.
    4 (b) (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 (a) Calculate the magnitude of the impulse. 5 (b) 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
  1. Use dimensional analysis to find the dimensions of \(k\) 6
  2. 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
      1. (ii) Find the value of \(e\) 7
    2. 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
  1. 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
  2. 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
  1. Find the value of \(\omega\) 4
  2. (i) Find the magnitude of the horizontal resultant force acting on Reena.
    4 (b) (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 (a) Calculate the magnitude of the impulse. 5 (b) 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 (a) Use dimensional analysis to find the dimensions of \(k\) 6 (b) 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 (a)
    1. Show that \(A\) does not change its direction of motion as a result of the collision.
      7
    2. Find the value of \(e\)
    7 (b) Given that the mass of \(B\) is 0.6 kg , find the mass of \(A\)
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|}{}23
21
31
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 }\(\mathbf { S } _ { \mathbf { 1 } }\)\(\mathbf { S } _ { \mathbf { 2 } }\)\(\mathbf { S } _ { \mathbf { 3 } }\)
\(\mathbf { A } _ { \mathbf { 1 } }\)223
\cline { 2 - 5 }\(\mathbf { A } _ { \mathbf { 2 } }\)035
\(\mathbf { A } _ { \mathbf { 3 } }\)- 12- 2
3
  1. Explain why the value of the game is 2
    3
  2. 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}