AQA Further AS Paper 2 Mechanics (Further AS Paper 2 Mechanics) 2024 June

1 An elastic string has modulus of elasticity 20 newtons and natural length 2 metres.
The string is stretched so that its extension is 0.5 metres.
Find the elastic potential energy stored in the string.
Circle your answer.
[0pt] [1 mark]
1.25 J
5.5 J
5 J
10 J

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

3 A cyclist travels around a circular track of radius 20 m at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the angular speed of the cyclist in radians per second. Circle your answer.
[0pt] [1 mark]
\(0.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(0.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(3.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\)

5 J
10 J 2 State the dimensions of impulse.
Circle your answer.
[0pt] [1 mark]
\(M L T ^ { - 2 }\)
\(M L T ^ { - 1 }\)
MLT
\(M L T { } ^ { 2 }\) 3 A cyclist travels around a circular track of radius 20 m at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the angular speed of the cyclist in radians per second. Circle your answer.
[0pt] [1 mark]
\(0.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(0.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(3.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\)

6 Kepler's Third Law of planetary motion for the period of a circular orbit around the Earth is given by the formula, $$t = 2 \pi \sqrt { \frac { r ^ { 3 } } { G m } }$$ where,
\(t\) is the time taken for one orbit
\(r\) is the radius of the circular orbit
\(m\) is the mass of the Earth
\(G\) is a gravitational constant. Use dimensional analysis to determine the dimensions of \(G\)
\includegraphics[max width=\textwidth, alt={}, center]{ce05dedd-515b-49e2-92f3-f5ec22bab4be-08_2491_1755_173_123}

7 A single force, \(F\) newtons, acts on a particle moving on a straight, smooth, horizontal line. The force \(F\) acts in the direction of motion of the particle.
At time \(t\) seconds, \(F = 6 \mathrm { e } ^ { t } + 2 \mathrm { e } ^ { 2 t }\) where \(0 \leq t \leq \ln 8\) 7

1. Find the impulse of \(F\) over the interval \(0 \leq t \leq \ln 8\)
   7
2. The particle has a mass of 2 kg and at time \(t = 0\) has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the velocity of the particle when \(t = \ln 8\)

8 Two spheres, \(A\) and \(B\), of equal size are moving in the same direction along a straight line on a smooth horizontal surface. Sphere \(A\) has mass \(m\) and is moving with speed \(4 u\) Sphere \(B\) has mass \(6 m\) and is moving with speed \(u\)
The diagram shows the spheres and their velocities. \begin{figure}[h] \end{figure} B Subsequently \(A\) collides directly with \(B\) The coefficient of restitution between \(A\) and \(B\) is \(e\) 8

1. Find, in terms of \(m\) and \(u\), the total momentum of the spheres before the collision.
   8
2. Show that the speed of \(B\) immediately after the collision is \(\frac { u ( 3 e + 10 ) } { 7 }\)
   8
3. After the collision sphere \(A\) moves in the opposite direction.
   Find the range of possible values for \(e\)
   [0pt] [5 marks]

10 J 2 State the dimensions of impulse.
Circle your answer.
[0pt] [1 mark]
\(M L T ^ { - 2 }\)
\(M L T ^ { - 1 }\)
MLT
\(M L T { } ^ { 2 }\) 3 A cyclist travels around a circular track of radius 20 m at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the angular speed of the cyclist in radians per second. Circle your answer.
[0pt] [1 mark]
\(0.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(0.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\)
\(3.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\)