AQA Further AS Paper 2 Mechanics (Further AS Paper 2 Mechanics) 2021 June

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

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

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]

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]

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]

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]

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]

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. 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]
3. 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]