6 A section of a golf practice ground is inclined at $15 ^ { \circ }$ to the horizontal. A golfer is hitting a ball up and down a line of greatest slope of this section of the practice ground.

The golfer hits the ball up the slope, so that the ball initially makes an angle of $30 ^ { \circ }$ with the slope. The ball first bounces on the slope 50 m from its point of projection.
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\item Determine the initial speed of the ball.

The golfer now hits the ball down the slope. The ball initially moves with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the ball initially travels at an angle $\theta$ above the horizontal, as shown in Fig. 6. The ball first bounces at a point a distance $L$ m down the slope.

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  \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-6_545_791_794_242}
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\caption{Fig. 6}
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\item Show that $\mathrm { L } = \frac { 800 } { \mathrm {~g} } \left( \frac { \sin \theta \cos \theta } { \cos 15 ^ { \circ } } + \frac { \sin 15 ^ { \circ } \cos ^ { 2 } \theta } { \cos ^ { 2 } 15 ^ { \circ } } \right)$.

You are given that $\frac { \mathrm { dL } } { \mathrm { d } \theta } = \frac { 800 } { \mathrm {~g} } \left( \frac { \cos 2 \theta } { \cos 15 ^ { \circ } } - \frac { \sin 15 ^ { \circ } \sin 2 \theta } { \cos ^ { 2 } 15 ^ { \circ } } \right)$.
\item Determine the value of $\theta$ for which $\frac { \mathrm { d } L } { \mathrm {~d} \theta } = 0$.
\item Hence determine the maximum distance the golfer can hit the ball down the slope.
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\hfill \mbox{\textit{OCR MEI Further Mechanics B AS 2021 Q6 [13]}}