14 J\\
18J\\
42 J

3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days.

Model this orbit as circular, with a radius of $3.84 \times 10 ^ { 8 }$ metres.\\
Find the approximate speed of the moon relative to Earth, in metres per second.\\

4 A particle $P$, of mass $m \mathrm {~kg}$, collides with a particle $Q$, of mass 2 kg

Immediately before the collision the velocity of $P$ is $\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$

As a result of the collision the particles coalesce into a single particle which moves with velocity $\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant.

Find the value of $k$.\\

5 A train consisting of an engine and eight carriages moves on a straight horizontal track.

A constant resistive force of 2400 N acts on the engine.\\
A constant resistive force of 300 N acts on each of the eight carriages.\\
The maximum speed of the train on the track is $120 \mathrm {~km} \mathrm {~h} ^ { - 1 }$\\
Find the maximum power output of the engine.\\
Fully justify your answer.\\

6 The magnitude of the gravitational force $F$ between two planets of masses $m _ { 1 }$ and $m _ { 2 }$ with centres at a distance $d$ apart is given by

$$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$

where $G$ is a constant.\\
6
\begin{enumerate}[label=(\alph*)]
\item Show that $G$ must have dimensions $L ^ { 3 } M ^ { - 1 } T ^ { - 2 }$, where $L$ represents length, $M$ represents mass and $T$ represents time.\\

6
\item The lifetime $t$ of a planet is thought to depend on its mass $m$, its radius $r$, the constant $G$ and a dimensionless constant $k$ such that

$$t = k m ^ { a } r ^ { b } G ^ { c }$$

where $a , b$ and $c$ are constants.\\
Determine the values of $a , b$ and $c$.\\

7 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

As part of a competition, Jo-Jo makes a small pop-up rocket.\\
It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket.

The rocket is released from rest and moves vertically upwards.\\
The mass of the rocket is 18 grams and the stiffness constant of the spring is $60 \mathrm { Nm } ^ { - 1 }$

Initially the spring is compressed by 3 cm\\
7 (a) Find the speed of the rocket when the spring first reaches its natural length.\\

7 (b) By considering energy find the distance that the rocket rises.

7
\item In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position.

With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not.

Fully justify your answer.\\

8 Two smooth spheres $A$ and $B$ have the same radius and are free to move on a smooth horizontal surface.

The masses of $A$ and $B$ are $2 m$ and $m$ respectively.\\
Both $A$ and $B$ are initially at rest.\\
The sphere $A$ is set in motion directly towards $B$ with speed $3 u$ and at the same time $B$ is set in motion directly towards $A$ with speed $2 u$.

Subsequently $A$ and $B$ collide directly.

$A$

The coefficient of restitution between the spheres is $e$.\\
8 (a) Show that the speed of $B$ after the collision is given by

$$\frac { 2 u ( 2 + 5 e ) } { 3 }$$

\section*{Question 8 continues on the next page}
8 (b) Given that the direction of the velocity of $A$ is reversed during the collision, find the range of possible values of $e$.

Fully justify your answer.\\[0pt]
[4 marks]\\

8 (c) Given that the magnitude of the impulse that $A$ exerts on $B$ is $\frac { 19 m u } { 3 }$, find the value of $e$.\\

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\hfill \mbox{\textit{AQA Further AS Paper 2 Mechanics 2020 Q14 [4]}}