6 Two uniform smooth spheres $A$ and $B$ of equal radius are moving on a smooth horizontal surface when they collide. $A$ has mass 2.5 kg and $B$ has mass 3 kg . Immediately before the collision $A$ and $B$ each has speed $u \mathrm {~ms} ^ { - 1 }$ and each moves in a direction at an angle $\theta$ to their line of centres, as indicated in Fig. 1. Immediately after the collision $A$ has speed $v _ { 1 } \mathrm {~ms} ^ { - 1 }$ and moves in a direction at an angle $\alpha$ to the line of centres, and $B$ has speed $v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and moves in a direction at an angle $\beta$ to the line of centres as indicated in Fig. 2. The coefficient of restitution between $A$ and $B$ is $e$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-4_336_814_667_699}
\caption{Fig. 1}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-4_374_657_1228_767}
\caption{Fig. 2}
\end{center}
\end{figure}

(i) Show that $\tan \beta = \frac { 11 \tan \theta } { 10 e - 1 }$.\\
(ii) Given that after impact sphere $A$ moves at an angle of $50 ^ { \circ }$ to the line of centres and $B$ moves perpendicular to the line of centres, find $\theta$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-5_817_848_374_210}
\caption{Fig. 3}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-5_819_953_376_1062}
\caption{Fig. 4}
\end{center}
\end{figure}

The region bounded by the $x$-axis, the $y$-axis, the line $x = \ln 32$ and the curve $y = \mathrm { e } ^ { 0.8 x }$ for $0 \leq x \leq \ln 32$, is occupied by a uniform lamina (see Fig. 3).\\
(i) Show that the $x$-coordinate of the centre of mass of the lamina is given by $\frac { 16 } { 3 } \ln 2 - \frac { 5 } { 4 }$.\\
(ii) Calculate the $y$-coordinate of the centre of mass of the lamina.\\
(iii) The region bounded by the $x$-axis, the line $x = 16$ and the curve $y = 1.25 \ln x$ for $1 \leq x \leq 16$, is occupied by a second uniform lamina (see Fig. 4). By using your answer to part (i) find, to 3 significant figures, the $x$-coordinate of the centre of mass of this second lamina.

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\hfill \mbox{\textit{OCR Further Mechanics  Q6 [12]}}