AQA Further AS Paper 2 Mechanics (Further AS Paper 2 Mechanics) 2023 June

1 A particle moves along the \(x\)-axis under the action of a force, \(F\) newtons, where $$F = 3 x ^ { 2 } + 5$$ Find the work done by the force as the particle moves from \(x = 0\) metres to \(x = 2\) metres. Circle your answer.
12 J
17 J
18 J
34 J

2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer.
\(e = - 1\)
\(e = 0\)
\(e = \frac { 1 } { 2 }\)
\(e = 1\)

3 A stone of mass 0.2 kg is thrown vertically upwards with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Find the initial kinetic energy of the stone.
Circle your answer.
[0pt] [1 mark]
1 J
5 J
10 J
20 J

5 J
10 J
20 J 4 Reena is skating on an ice rink, which has a horizontal surface. She follows a circular path of radius 5 metres and centre \(O\)
She completes 10 full revolutions in 1 minute, moving with a constant angular speed of \(\omega\) radians per second. The mass of Reena is 40 kg
4

1. Find the value of \(\omega\)
   4
2. 1. Find the magnitude of the horizontal resultant force acting on Reena.
      4
3. (ii) Show the direction of this horizontal resultant force on the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-03_380_442_2017_861} 5 An impulse of \(\left[ \begin{array} { r } - 5
   12 \end{array} \right] \mathrm { N } \mathrm { s }\) is applied to a particle of mass 5 kg which is moving with velocity \(\left[ \begin{array} { l } 6
   2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) 5
4. Calculate the magnitude of the impulse. 5
5. Find the speed of the particle immediately after the impulse is applied.

6 A ball is thrown with speed \(u\) at an angle of \(45 ^ { \circ }\) to the horizontal from a point \(O\) When the horizontal displacement of the ball is \(x\), the vertical displacement of the ball above \(O\) is \(y\) where $$y = x - \frac { k x ^ { 2 } } { u ^ { 2 } }$$ 6

1. Use dimensional analysis to find the dimensions of \(k\)
   6
2. State what can be deduced about \(k\) from the dimensions that you found in part (a).

7 Two smooth, equally sized spheres, \(A\) and \(B\), are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
Immediately after the collision, \(A\) has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the spheres is \(e\) 7

1. 1. Show that \(A\) does not change its direction of motion as a result of the collision.
      7
2. (ii) Find the value of \(e\)
   7
3. Given that the mass of \(B\) is 0.6 kg , find the mass of \(A\)

8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Omar, a bungee jumper of mass 70 kg , has his ankles attached to one end of an elastic cord. The other end of the cord is attached to a bridge which is 80 metres above the surface of a river. Omar steps off the bridge at the point where the cord is attached and falls vertically downwards. The cord can be modelled as a light elastic string of natural length \(L\) metres and modulus of elasticity 2800 N Model Omar as a particle. 8

1. Given that Omar just reaches the surface of the river before being pulled back up, find the value of \(L\) Fully justify your answer.
   8
2. If Omar is not modelled as a particle, explain the effect of revising this assumption on your answer to part (a).

18 J
34 J 2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer.
\(e = - 1\)
\(e = 0\)
\(e = \frac { 1 } { 2 }\)
\(e = 1\) 3 A stone of mass 0.2 kg is thrown vertically upwards with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Find the initial kinetic energy of the stone.
Circle your answer.
[0pt] [1 mark]
1 J
5 J
10 J

20 J 4 Reena is skating on an ice rink, which has a horizontal surface. She follows a circular path of radius 5 metres and centre \(O\)
She completes 10 full revolutions in 1 minute, moving with a constant angular speed of \(\omega\) radians per second. The mass of Reena is 40 kg
4

1. Find the value of \(\omega\)
   4
2. 1. Find the magnitude of the horizontal resultant force acting on Reena.
      4
3. (ii) Show the direction of this horizontal resultant force on the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-03_380_442_2017_861} 5 An impulse of \(\left[ \begin{array} { r } - 5
   12 \end{array} \right] \mathrm { N } \mathrm { s }\) is applied to a particle of mass 5 kg which is moving with velocity \(\left[ \begin{array} { l } 6
   2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) 5
4. Calculate the magnitude of the impulse. 5
5. Find the speed of the particle immediately after the impulse is applied.
   6 A ball is thrown with speed \(u\) at an angle of \(45 ^ { \circ }\) to the horizontal from a point \(O\) When the horizontal displacement of the ball is \(x\), the vertical displacement of the ball above \(O\) is \(y\) where $$y = x - \frac { k x ^ { 2 } } { u ^ { 2 } }$$ 6
6. Use dimensional analysis to find the dimensions of \(k\)
   6
7. State what can be deduced about \(k\) from the dimensions that you found in part (a).
   7 Two smooth, equally sized spheres, \(A\) and \(B\), are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
   Immediately after the collision, \(A\) has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the spheres is \(e\) 7
8. 1. Show that \(A\) does not change its direction of motion as a result of the collision.
      7
9. (ii) Find the value of \(e\)
   7
10. Given that the mass of \(B\) is 0.6 kg , find the mass of \(A\)