A particle \(P\) is projected from a fixed point \(O\). The particle is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\alpha\) above the horizontal. The particle moves freely under gravity. At the instant when the horizontal distance of \(P\) from \(O\) is \(x\) metres, \(P\) is \(y\) metres vertically above the level of \(O\).
- Show that \(y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)\)
A small ball is projected from a fixed point \(A\) with speed \(U \mathrm {~ms} ^ { - 1 }\) at \(\theta ^ { \circ }\) above the horizontal.
The point \(B\) is on horizontal ground and is vertically below the point \(A\), with \(A B = 20 \mathrm {~m}\).
The ball hits the ground at the point \(C\), where \(B C = 30 \mathrm {~m}\), as shown in Figure 4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee5f81bc-1bdb-47a1-81e7-7e3cb8219e91-24_556_961_904_552}
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\caption{Figure 4}
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The speed of the ball immediately before it hits the ground is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The motion of the ball is modelled as that of a particle moving freely under gravity.