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]
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\caption{Fig. 1}
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\caption{Fig. 2}
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- Show that \(\tan \beta = \frac { 11 \tan \theta } { 10 e - 1 }\).
- 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]
\includegraphics[alt={},max width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-5_817_848_374_210}
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\caption{Fig. 3}
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\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-5_819_953_376_1062}
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\caption{Fig. 4}
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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). - Show that the \(x\)-coordinate of the centre of mass of the lamina is given by \(\frac { 16 } { 3 } \ln 2 - \frac { 5 } { 4 }\).
- Calculate the \(y\)-coordinate of the centre of mass of the lamina.
- 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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