- \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors, with \(\mathbf { i }\) horizontal and \(\mathbf { j }\) vertical.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-12_278_891_294_587}
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\caption{Figure 3}
\end{figure}
The fixed points \(A\) and \(B\) lie on horizontal ground.
At time \(t = 0\), a particle \(P\) is projected from \(A\) with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\)
Particle \(P\) moves freely under gravity and hits the ground at \(B\), as shown in Figure 3 .
At time \(T _ { 1 }\) seconds, \(P\) is at its highest point above the ground.
- Find the value of \(T _ { 1 }\)
At time \(t = 0\), a particle \(Q\) is also projected from \(A\) but with velocity \(( 5 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Particle \(Q\) moves freely under gravity.
- Find the vertical distance between \(Q\) and \(P\) at time \(T _ { 1 }\) seconds, giving your answer to 2 significant figures.
At the instant when particle \(P\) reaches \(B\), particle \(Q\) is moving at \(\alpha ^ { \circ }\) below the horizontal.
- Find the value of \(\alpha\).
At time \(T _ { 2 }\) seconds, the direction of motion of \(Q\) is perpendicular to the initial direction of motion of \(Q\).
- Find the value of \(T _ { 2 }\)