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

1 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 \(k \mathrm { Nm } ^ { - 1 }\)
Find \(k\) Circle your answer. 2050100200

2 State the dimensions of force. Circle your answer.
[0pt] [1 mark]
MLT
\(M L ^ { 2 } T\)
\(M L T ^ { - 1 }\)
\(M L T ^ { - 2 }\)

3 Use \(g\) as \(9.8 \mathrm {~ms} ^ { - 2 }\) 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 \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The mass of 1 litre of water is 1 kg
3

1. Find the gain in the potential energy of the 300 litres of water.
   3
2. \(\quad\) Calculate \(v\)

4 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.
4

1. Work out the greatest speed, in \(\mathrm { km } \mathrm { h } ^ { - 1 }\), at which the cyclist can travel around the bend.
   4
2. With reference to the surface of the road, describe one limitation of the model.
   \(5 \quad\) A ball is thrown vertically upwards with speed \(u\) so that at time \(t\) its displacement \(s\) is given by the formula $$s = u t - \frac { g t ^ { 2 } } { 2 }$$ Use dimensional analysis to show that this formula is dimensionally consistent.
   Fully justify your answer.
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6 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 \(28 \mathrm {~ms} ^ { - 1 }\) Immediately after the impact, the ball is travelling horizontally with speed \(14 \mathrm {~ms} ^ { - 1 }\) in the opposite direction. 6

1. Find the magnitude of the impulse exerted by the bat on the ball.
   [0pt] [2 marks]
   6
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 $$F = 10 k t ( 0.05 - t )$$ where \(k\) is a constant.
   Given the ball is in contact with the bat for 0.05 seconds, find the value of \(k\)
   [0pt] [3 marks]
   \(7 \quad\) Use \(g\) as \(9.81 \mathrm {~ms} ^ { - 2 }\) in this question. A light elastic string has one end attached to a fixed point \(A\) on a smooth plane inclined at \(25 ^ { \circ }\) 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.

7

1. Find the elastic potential energy of the string at the point when the block is released.
   7
2. Calculate the speed of the block when the string becomes slack.
   7
3. Determine whether the block reaches the point \(A\) in the subsequent motion, commenting on any assumptions that you make.

8 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 \(4 \mathrm {~ms} ^ { - 1 }\) and subsequently collides with sphere \(B\) The coefficient of restitution between the spheres is \(e\)
8

1. 1. Show that the speed of \(B\) immediately after the collision is $$\frac { 8 ( 1 + e ) } { 5 }$$ 8
2. (ii) Find an expression, in terms of \(e\), for the velocity of \(A\) immediately after the collision.
   8
3. It is given that the spheres both move in the same direction after the collision. Find the range of possible values of \(e\)
   [0pt] [2 marks]
   8
4. 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 \(I + J\)
      8
5. (ii) State, giving a reason for your answer, whether the value found in part (c)(i) would change if the collision was not perfectly inelastic.
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