AQA Further AS Paper 2 Mechanics (Further AS Paper 2 Mechanics) 2022 June

1 A box is being pushed in a straight line along horizontal ground by a force.
The force is applied in the direction of motion and has magnitude 10 newtons. The box moves 5 metres in 2 seconds. Calculate the work done by the force.
Circle your answer.
20 J
25 J
50 J
100 J

2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2
4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6
- 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark]
\(\left[ \begin{array} { l } 4
1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 4
3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8
2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8
6 \end{array} \right] \mathrm { ms } ^ { - 1 }\)

3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3

1. Calculate the initial kinetic energy of the ball. 3
2. The maximum height of the ball above the ground is \(h\) metres.
   Jeff and Gurjas use an energy method to find \(h\)
   Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\)
   Explain the reasoning that they have used, showing any calculations that you make.

4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
Two of these classifications are deep water waves and shallow water waves.
4

1. The wavelength \(w\) of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant. 4
2. The wavelength \(w\) of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves. Use dimensional analysis to find the values of \(\alpha\) and \(\beta\)

5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
Find the maximum power output of the car.
Fully justify your answer.

6 An ice hockey puck, of mass 0.2 kg , is moving in a straight line on a horizontal ice rink under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude ( \(2 t + 3\) ) newtons.
The force acts on the puck from \(t = 0\) to \(t = T\)
6

1. Show that the magnitude of the impulse of the force is \(a T ^ { 2 } + b T\), where \(a\) and \(b\) are integers to be found.
   [0pt] [3 marks]
   6
2. While the force acts on the puck, its speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Use your answer from part (a) to find \(T\), giving your answer to three significant figures.
   Fully justify your answer.

7 The particles \(A\) and \(B\) are moving on a smooth horizontal surface directly towards each other. Particle \(A\) has mass 0.4 kg and particle \(B\) has mass 0.2 kg
Particle \(A\) has speed \(4 \mathrm {~ms} ^ { - 1 }\) and particle \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) when they collide, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{ec39a757-5867-4798-b26c-73cd5746581c-08_392_1064_625_488} The coefficient of restitution between the particles is \(e\)
7

1. Find the magnitude of the total momentum of the particles before the collision.
   [0pt] [2 marks] 7
2. 1. Show that the speed of \(B\) immediately after the collision is \(( 4 e + 2 ) \mathrm { ms } ^ { - 1 }\)
      [0pt] [3 marks]
      7
3. (ii) Find an expression, in terms of \(e\), for the speed of \(A\) immediately after the collision.
   7
4. Explain what happens to particle \(A\) when the collision is perfectly elastic.

20 J
25 J
50 J
100 J 2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2
4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6
- 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark]
\(\left[ \begin{array} { l } 4
1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 4
3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8
2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8
6 \end{array} \right] \mathrm { ms } ^ { - 1 }\) 3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3

1. Calculate the initial kinetic energy of the ball. 3
2. The maximum height of the ball above the ground is \(h\) metres.
   Jeff and Gurjas use an energy method to find \(h\)
   Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\)
   Explain the reasoning that they have used, showing any calculations that you make.
   4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
   Two of these classifications are deep water waves and shallow water waves.
   4
3. The wavelength \(w\) of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant. 4
4. The wavelength \(w\) of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves. Use dimensional analysis to find the values of \(\alpha\) and \(\beta\)
   5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
   Find the maximum power output of the car.
   Fully justify your answer.