AQA Further AS Paper 2 Mechanics (Further AS Paper 2 Mechanics) 2018 June

1 A particle \(A\), of mass 0.2 kg , collides with a particle \(B\), of mass 0.3 kg Immediately before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 4
12 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
and the velocity of \(B\) is \(\left[ \begin{array} { l } - 1
- 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle.
Circle your answer.
[0pt] [1 mark] $$\left[ \begin{array} { l } 0.5
1.5 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 2
6 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 1
3 \end{array} \right] \mathrm { ms } ^ { - 1 } \quad \left[ \begin{array} { l } 3
9 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 }$$

2 A train is travelling at maximum speed with its engine using its maximum power of 1800 kW When travelling at this speed the train experiences a total resistive force of 40000 N Find the maximum speed of the train. Circle your answer.
[0pt] [1 mark]
\(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
\(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
\(54 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
\(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

3 The kinetic energy, \(E\), of a compound pendulum is given by $$E = \frac { 1 } { 2 } I \omega ^ { 2 }$$ where \(\omega\) is the angular speed and \(I\) is a quantity called the moment of inertia.
3

1. Show that for this formula to be dimensionally consistent then \(I\) must have dimensions \(M L ^ { 2 }\), where \(M\) represents mass and \(L\) represents length.
   [0pt] [2 marks]
   3
2. The time, \(T\), taken for one complete swing of a pendulum is thought to depend on its moment of inertia, \(I\), its weight, \(W\), and the distance, \(h\), of the centre of mass of the pendulum from the point of suspension. The formula being proposed is $$T = k I ^ { \alpha } W ^ { \beta } h ^ { \gamma }$$ where \(k\) is a dimensionless constant. Determine the values of \(\alpha , \beta\) and \(\gamma\).

4 Two smooth spheres \(A\) and \(B\) of equal radius are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(m\) and \(4 m\) respectively.
The coefficient of restitution between the spheres is \(e\).
The spheres are projected directly towards each other, each with speed \(u\), and subsequently collide. 4

1. Show that the speed of \(B\) immediately after the impact with \(A\) is $$\frac { u ( 3 - 2 e ) } { 5 }$$ 4
2. Find the speed of \(A\) in terms of \(u\) and \(e\).
   4
3. Comment on the direction of motion of the spheres after the collision, justifying your answer.
   4
4. The magnitude of the impulse on \(B\) due to the collision is \(I\).
   Deduce that $$\frac { 8 m u } { 5 } \leq I \leq \frac { 16 m u } { 5 }$$

5 A car travels around a roundabout at a constant speed. The surface of the roundabout is horizontal. The car has mass 990 kg and the path of the car is a circular arc of radius 48 metres.
A simple model assumes that the car is a particle and the only horizontal force acting on it as it travels around the roundabout is friction. On a dry day typical values of friction, \(F\), between the surface of the roundabout and the tyres of the car are $$7300 \mathrm {~N} \leq F \leq 9200 \mathrm {~N}$$ 5

1. Using this model calculate a safe speed limit, in miles per hour, for the car as it travels around the roundabout. Explain your reasoning fully.
   Note that there are 1600 metres in one mile.
   5
2. Gary assumes that on a wet day typical values for friction, \(F\), are $$5400 \mathrm {~N} \leq F \leq 10000 \mathrm {~N}$$ Comment on the validity of Gary's revised assumption.

6 At a fairground a dodgem car is moving in a straight horizontal line towards a side wall that is perpendicular to the velocity of the car. The speed of the car is \(1.8 \mathrm {~ms} ^ { - 1 }\)
It collides with the side wall and rebounds along its original path with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total mass of the dodgem car and the passengers is 250 kg
6

1. Find the magnitude of the impulse on the car during the collision with the side wall.
   6
2. A possible model for the magnitude of the force, \(F\) newtons, acting on the dodgem car due to its collision with the side wall is given by $$F = k t ( 4 - 5 t ) \quad \text { for } 0 \leq t \leq 0.8$$ 6
3. 1. Find the value of \(k\).
4. (ii) Determine the maximum magnitude of the force predicted by the model. 6
5. (ii) Determine the maximum magnitude of the fored bed bed at

7

1. Find Dominic's speed at the point when the cord initially becomes taut.
   7
2. Determine whether or not Dominic enters the river and gets wet.
   7
3. One limitation of this model is that Dominic is not a particle.
   Explain the effect of revising this assumption on your answer to part (b).
   \includegraphics[max width=\textwidth, alt={}, center]{1b79a789-c003-46c9-9235-254c1d8a0501-12_2492_1721_217_150} Question number Additional page, if required.
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