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AQA Further AS Paper 2 Mechanics 2019 June Q1
1 marks Easy -1.8
A turntable rotates at a constant speed of \(33\frac{1}{3}\) revolutions per minute. Find the angular speed in radians per second. Circle your answer. [1 mark] \(\frac{5\pi}{9}\) \quad \(\frac{10\pi}{9}\) \quad \(\frac{5\pi}{3}\) \quad \(\frac{20\pi}{9}\)
AQA Further AS Paper 2 Mechanics 2019 June Q2
1 marks Easy -1.2
The graph shows the resistance force experienced by a cyclist over the first 20 metres of a bicycle ride. \includegraphics{figure_2} Find the work done by the resistance force over the 20 metres of the bicycle ride. Circle your answer. [1 mark] 1600 J \quad 3000 J \quad 3200 J \quad 4000 J
AQA Further AS Paper 2 Mechanics 2019 June Q3
3 marks Moderate -0.5
A formula for the elastic potential energy, \(E\), stored in a stretched spring is given by $$E = \frac{kx^2}{2}$$ where \(x\) is the extension of the spring and \(k\) is a constant. Use dimensional analysis to find the dimensions of \(k\). [3 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q4
7 marks Standard +0.3
In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) A ride in a fairground consists of a hollow vertical cylinder of radius 4.6 metres with a horizontal floor. Stephi, who has mass 50 kilograms, stands inside the cylinder with her back against the curved surface. The cylinder begins to rotate about a vertical axis through the centre of the cylinder. When the cylinder is rotating at a constant angular speed of \(\omega\) radians per second, the magnitude of the normal reaction between Stephi and the curved surface is 980 newtons. The floor is lowered and Stephi remains against the curved surface with her feet above the floor, as shown in the diagram. \includegraphics{figure_4}
  1. Explain, with the aid of a force diagram, why the magnitude of the frictional force acting on Stephi is 490 newtons. [2 marks]
  2. Find \(\omega\) [3 marks]
  3. State one modelling assumption that you have used in this question. Explain the effect of this assumption. [2 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q5
7 marks Standard +0.8
A car of mass 1000 kg has a maximum speed of \(40\,\text{m}\,\text{s}^{-1}\) when travelling on a straight horizontal race track. The maximum power output of the car's engine is 48 kW The total resistance force experienced by the car can be modelled as being proportional to the car's speed. Find the maximum possible acceleration of the car when it is travelling at \(25\,\text{m}\,\text{s}^{-1}\) on the straight horizontal race track. Fully justify your answer. [7 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q6
9 marks Standard +0.3
In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) Martin, who is of mass 40 kg, is using a slide. The slide is made of two straight sections \(AB\) and \(BC\). The section \(AB\) has length 15 metres and is at an angle of \(50°\) to the horizontal. The section \(BC\) has length 2 metres and is horizontal. \includegraphics{figure_6} Martin pushes himself from \(A\) down the slide with initial speed \(1\,\text{m}\,\text{s}^{-1}\) He reaches \(B\) with speed \(5\,\text{m}\,\text{s}^{-1}\) Model Martin as a particle.
  1. Find the energy lost as Martin slides from \(A\) to \(B\). [4 marks]
  2. Assume that a resistance force of constant magnitude acts on Martin while he is moving on the slide.
    1. Show that the magnitude of this resistance force is approximately 270 N [2 marks]
    2. Determine if Martin reaches the point \(C\). [3 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q7
12 marks Standard +0.3
Two smooth spheres, \(P\) and \(Q\), of equal radius are free to move on a smooth horizontal surface. The masses of \(P\) and \(Q\) are \(3m\) and \(m\) respectively. \(P\) is set in motion with speed \(u\) directly towards \(Q\), which is initially at rest. \(P\) subsequently collides with \(Q\). \includegraphics{figure_7} Immediately after the collision, \(P\) moves with speed \(v\) and \(Q\) moves with speed \(w\). The coefficient of restitution between the spheres is \(e\).
    1. Show that $$v = \frac{u(3-e)}{4}$$ [4 marks]
    2. Find \(w\), in terms of \(e\) and \(u\), simplifying your answer. [2 marks]
  2. Deduce that $$\frac{u}{2} \leq v \leq \frac{3u}{4}$$ [2 marks]
    1. Find, in terms of \(m\) and \(u\), the maximum magnitude of the impulse that \(P\) exerts on \(Q\). [3 marks]
    2. Describe the impulse that \(Q\) exerts on \(P\). [1 mark]
AQA Further AS Paper 2 Mechanics 2021 June Q1
1 marks Easy -1.8
A light spring of natural length 0.6 metres is compressed to a length of 0.4 metres by a force of 20 newtons. The stiffness of the spring is \(k\) N m\(^{-1}\) Find \(k\) Circle your answer. [1 mark] 20 50 100 200
AQA Further AS Paper 2 Mechanics 2021 June Q2
1 marks Easy -2.0
State the dimensions of force. Circle your answer. [1 mark] \(MLT\) \(ML^2T\) \(MLT^{-1}\) \(MLT^{-2}\)
AQA Further AS Paper 2 Mechanics 2021 June Q3
5 marks Standard +0.3
Use \(g\) as 9.8 m s\(^{-2}\) in this question. A pump is used to pump water out of a pool. The pump raises the water through a vertical distance of 5 metres and then ejects it through a pipe. The pump works at a constant rate of 400 W Over a period of 50 seconds, 300 litres of water are pumped out of the pool and the water is ejected with speed \(v\) m s\(^{-1}\) The mass of 1 litre of water is 1 kg
  1. Find the gain in the potential energy of the 300 litres of water. [1 mark]
  2. Calculate \(v\) [4 marks]
AQA Further AS Paper 2 Mechanics 2021 June Q4
5 marks Standard +0.3
A cyclist in a road race is travelling around a bend on a horizontal circular path of radius 15 metres and is prevented from skidding by a frictional force. The frictional force has a maximum value of 500 newtons. The total mass of the cyclist and his cycle is 75 kg Assume that the cyclist travels at a constant speed.
  1. Work out the greatest speed, in km h\(^{-1}\), at which the cyclist can travel around the bend. [4 marks]
  2. With reference to the surface of the road, describe one limitation of the model. [1 mark]
AQA Further AS Paper 2 Mechanics 2021 June Q5
4 marks Moderate -0.8
A ball is thrown vertically upwards with speed \(u\) so that at time \(t\) its displacement \(s\) is given by the formula $$s = ut - \frac{gt^2}{2}$$ Use dimensional analysis to show that this formula is dimensionally consistent. Fully justify your answer. [4 marks]
AQA Further AS Paper 2 Mechanics 2021 June Q6
5 marks Moderate -0.3
A ball of mass 0.15 kg is hit directly by a vertical cricket bat. Immediately before the impact, the ball is travelling horizontally with speed 28 m s\(^{-1}\) Immediately after the impact, the ball is travelling horizontally with speed 14 m s\(^{-1}\) in the opposite direction.
  1. Find the magnitude of the impulse exerted by the bat on the ball. [2 marks]
  2. In a simple model the force, \(F\) newtons, exerted by the bat on the ball, \(t\) seconds after the initial impact, is given by \(F = 10kt (0.05 - t)\) where \(k\) is a constant. Given the ball is in contact with the bat for 0.05 seconds, find the value of \(k\) [3 marks]
AQA Further AS Paper 2 Mechanics 2021 June Q7
8 marks Challenging +1.2
Use \(g\) as 9.81 m s\(^{-2}\) in this question. A light elastic string has one end attached to a fixed point A on a smooth plane inclined at 25° to the horizontal. The other end of the string is attached to a wooden block of mass 2.5 kg, which rests on the plane. The elastic string has natural length 3 metres and modulus of elasticity 125 newtons. The block is pulled down the line of greatest slope of the plane to a point 4.5 metres from A and then released.
  1. Find the elastic potential energy of the string at the point when the block is released. [1 mark]
  2. Calculate the speed of the block when the string becomes slack. [4 marks]
  3. Determine whether the block reaches the point A in the subsequent motion, commenting on any assumptions that you make. [3 marks]
AQA Further AS Paper 2 Mechanics 2021 June Q8
11 marks Standard +0.3
Two spheres A and B are free to move on a smooth horizontal surface. The masses of A and B are 2 kg and 3 kg respectively. Both A and B are initially at rest. Sphere A is set in motion directly towards sphere B with speed 4 m s\(^{-1}\) and subsequently collides with sphere B The coefficient of restitution between the spheres is \(e\)
    1. Show that the speed of B immediately after the collision is $$\frac{8(1 + e)}{5}$$ [4 marks]
    2. Find an expression, in terms of \(e\), for the velocity of A immediately after the collision. [2 marks]
  2. It is given that the spheres both move in the same direction after the collision. Find the range of possible values of \(e\) [2 marks]
    1. The impulse of sphere A on sphere B is \(I\) The impulse of sphere B on sphere A is \(J\) Given that the collision is perfectly inelastic, find the value of \(I + J\) [1 mark]
    2. State, giving a reason for your answer, whether the value found in part (c)(i) would change if the collision was not perfectly inelastic. [2 marks]
AQA Further AS Paper 2 Mechanics 2024 June Q1
1 marks Easy -1.2
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. 1.25 J \quad\quad 5.5 J \quad\quad 5 J \quad\quad 10 J [1 mark]
AQA Further AS Paper 2 Mechanics 2024 June Q2
1 marks Easy -1.8
State the dimensions of impulse. Circle your answer. \(MLT^{-2}\) \quad\quad \(MLT^{-1}\) \quad\quad \(MLT\) \quad\quad \(MLT^2\) [1 mark]
AQA Further AS Paper 2 Mechanics 2024 June Q3
1 marks Easy -1.8
A cyclist travels around a circular track of radius 20 m at a constant speed of \(8 \text{ m s}^{-1}\) Find the angular speed of the cyclist in radians per second. Circle your answer. \(0.2 \text{ rad s}^{-1}\) \quad\quad \(0.4 \text{ rad s}^{-1}\) \quad\quad \(2.5 \text{ rad s}^{-1}\) \quad\quad \(3.2 \text{ rad s}^{-1}\) [1 mark]
AQA Further AS Paper 2 Mechanics 2024 June Q4
8 marks Standard +0.3
In this question use \(g = 9.8 \text{ m s}^{-2}\) A ball of mass 0.5 kg is projected vertically upwards with a speed of \(10 \text{ m s}^{-1}\)
  1. Calculate the initial kinetic energy of the ball. [1 mark]
  2. Assuming that the weight is the only force acting on the ball, use an energy method to show that the maximum height reached by the ball is approximately 5.1 m above the point of projection. [2 marks]
    1. A student conducts an experiment to verify the accuracy of the result obtained in part (b). They observe that the ball rises to a height of 4.4 m above the point of projection and concludes that this height difference is due to a resistance force, \(R\) newtons. Find the total work done against \(R\) whilst the ball is moving upwards. [2 marks]
    2. Using a model that assumes \(R\) is constant, find the magnitude of \(R\) [2 marks]
    3. Comment on the validity of the model used in part (c)(ii). [1 mark]
AQA Further AS Paper 2 Mechanics 2024 June Q5
4 marks Standard +0.3
Kang is riding a motorbike along a straight, horizontal road. The motorbike has a maximum power of 75 000 W The maximum speed of the motorbike is \(50 \text{ m s}^{-1}\) When the speed of the motorbike is \(v \text{ m s}^{-1}\), the resistance force is \(kv\) newtons. Find the value of \(k\) Fully justify your answer. [4 marks]
AQA Further AS Paper 2 Mechanics 2024 June Q6
4 marks Standard +0.3
Kepler's Third Law of planetary motion for the period of a circular orbit around the Earth is given by the formula, $$t = 2\pi\sqrt{\frac{r^3}{Gm}}$$ 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 \(G\) [4 marks]
AQA Further AS Paper 2 Mechanics 2024 June Q7
5 marks Standard +0.3
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, \(F = 6e^t + 2e^{2t}\) where \(0 \leq t \leq \ln 8\)
  1. Find the impulse of \(F\) over the interval \(0 \leq t \leq \ln 8\) [2 marks]
  2. The particle has a mass of 2 kg and at time \(t = 0\) has velocity \(5 \text{ m s}^{-1}\) Find the velocity of the particle when \(t = \ln 8\) [3 marks]
AQA Further AS Paper 2 Mechanics 2024 June Q8
10 marks Standard +0.3
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 \(4u\) Sphere \(B\) has mass \(6m\) and is moving with speed \(u\) The diagram shows the spheres and their velocities. \includegraphics{figure_8} Subsequently \(A\) collides directly with \(B\) The coefficient of restitution between \(A\) and \(B\) is \(e\)
  1. Find, in terms of \(m\) and \(u\), the total momentum of the spheres before the collision. [1 mark]
  2. Show that the speed of \(B\) immediately after the collision is \(\frac{u(3e + 10)}{7}\) [4 marks]
  3. After the collision sphere \(A\) moves in the opposite direction. Find the range of possible values for \(e\) [5 marks]
AQA Further AS Paper 2 Mechanics 2024 June Q9
6 marks Standard +0.3
A small coin is placed at a point \(C\) on a rough horizontal turntable, with centre \(O\), as shown in the diagram below. \includegraphics{figure_9} The mass of the coin is 3.6 grams. The distance \(OC\) is 20 cm The turntable rotates about a vertical axis through \(O\), with constant angular speed \(\omega\) radians per second.
  1. Draw a diagram to show all the forces acting on the coin. [1 mark]
  2. The maximum value of friction is 0.01 newtons and the coin does not slip during the motion. Find the maximum value of \(\omega\) Give your answer to two significant figures. [4 marks]
  3. State one modelling assumption you have made to answer part (b). [1 mark]
AQA Further AS Paper 2 Discrete 2021 June Q1
2 marks Easy -1.8
  1. A project consists of three activities \(A\), \(B\) and \(C\) An activity network for the project is shown in the diagram below. \includegraphics{figure_1} Find the value of \(x\) Circle your answer. [1 mark] 5 \quad 7 \quad 8 \quad 12
  2. Find the value of \(y\) Circle your answer. [1 mark] 5 \quad 7 \quad 8 \quad 15