AQA Further AS Paper 2 Mechanics (Further AS Paper 2 Mechanics) Specimen

1 A child, of mass 40 kg , moves at constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a fairground ride.
The path of the child is a circle of radius 4 metres.
Find the magnitude of the resultant force acting on the child.
Circle your answer.
[0pt] [1 mark]
6.3 N
50 N
130 N
250 N

2 The graph shows how a force, \(F\), varies with time during a period of 0.8 seconds.
\includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-03_440_960_568_516} Find the magnitude of the impulse of \(F\) during the 0.8 seconds.
Circle your answer.
[0pt] [1 mark]
1.0 Ns
1.6 Ns
2.2 Ns
3.2 Ns Turn over for the next question

3 A tank full of liquid has a hole made in its base.
Two students, Sarah and David, propose two different models for the speed, \(v\), at which liquid exits the tank. David thinks that \(v\) will depend on the height of the liquid in the tank, \(h\), the acceleration due to gravity, \(g\), and the density of the liquid, \(\rho\), such that \(v \propto g ^ { a } h ^ { b } \rho ^ { c }\) where \(a\), \(b\) and \(c\) are constants. Sarah thinks that \(v\) will not depend on the density of the liquid and suggests the model \(v \propto g ^ { a } h ^ { b }\) 3

1. By considering dimensions, explain which student's model should be rejected.
   [0pt] [2 marks]
   3
2. Find the values of the constants in order for the model that you did not reject in part (a) to be dimensionally consistent.
   [0pt] [2 marks]

4 A cricket ball of mass 156 grams is thrown from a point which is 1.5 metres above the ground, with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) A tennis ball of mass 58 grams is thrown from the same point, with the same speed.
Prove that both balls hit the ground with the same speed.
Clearly state any assumptions you have made and how you have used them.
[0pt] [5 marks]

5 Two small smooth discs, \(C\) and \(D\), have equal radii and masses of 2 kg and 3 kg respectively. The discs are sliding on a smooth horizontal surface towards each other and collide directly. Disc \(C\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and disc \(D\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as they collide. The coefficient of restitution between \(C\) and \(D\) is 0.6 The diagram shows the discs, viewed from above, before the collision.
\includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-06_343_712_868_753} 5

1. Show that the speed of \(D\) immediately after the collision is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 2 significant figures.
   5
2. Find the speed of \(C\) immediately after the collision.
   [0pt] [2 marks]
   5
3. In fact the horizontal surface on which the discs are sliding is not smooth.
   Explain how the introduction of friction will affect your answer to part (b).
   [0pt] [2 marks]
   Turn over for the next question

6 A car, of mass 1200 kg , moves on a straight horizontal road where it has a maximum speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) When the car travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences a resistance force which can be modelled as being of magnitude 30 v newtons. 6

1. Show that the power output of the car is 48000 W , when it is travelling at its maximum speed. 6
2. Find the maximum acceleration of the car when it is travelling at a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
   [0pt] [4 marks]

7 A disc, of mass 0.15 kg , slides across a smooth horizontal table and collides with a vertical wall which is perpendicular to the path of the disc. The disc is in contact with the wall for 0.02 seconds and then rebounds.
A possible model for the force, \(F\) newtons, exerted on the disc by the wall, whilst in contact, is given by $$F = k t ^ { 2 } ( t - b ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.020$$ where \(k\) and \(b\) are constants.
The force is initially zero and becomes zero again as the disc loses contact with the wall. 7

1. State the value of \(b\).
   7
2. Find the magnitude of the impulse on the disc, giving your answer in terms of \(k\).
   7
3. The disc is travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the wall.
   The disc rebounds with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
   Find \(k\).
   [0pt] [3 marks]

8 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A particle, of mass 2 kg , is attached to one end of a light elastic string of natural length 0.2 metres. The other end of the string is attached to a fixed point \(O\).
The particle is pulled down and released from rest at a point 0.6 metres directly below \(O\).
The particle then moves vertically and next comes to rest when it is 0.1 metres below \(O\).
Assume that no air resistance acts on the particle.
8

1. Find the maximum speed of the particle.
   [0pt] [6 marks]
   8
2. Describe one way in which the model you have used could be refined.