AQA Further AS Paper 2 Mechanics (Further AS Paper 2 Mechanics) 2024 June

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]

State the dimensions of impulse. Circle your answer. \(MLT^{-2}\) \quad\quad \(MLT^{-1}\) \quad\quad \(MLT\) \quad\quad \(MLT^2\) [1 mark]

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]

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]
3. 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]

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]

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]

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]

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]

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]