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\includegraphics[max width=\textwidth, alt={}, center]{8e1225a2-cb98-4b71-a4af-0150f093f852-3_698_1047_1297_550} A particle \(P\) is projected with speed \(V _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 1 }\) from a point \(O\) on horizontal ground. When \(P\) is vertically above a point \(A\) on the ground its height is 250 m and its velocity components are \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards (see diagram).

1. Show that \(V _ { 1 } = 86.0\) and \(\theta _ { 1 } = 62.3 ^ { \circ }\), correct to 3 significant figures. At the instant when \(P\) is vertically above \(A\), a second particle \(Q\) is projected from \(O\) with speed \(V _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 2 } . P\) and \(Q\) hit the ground at the same time and at the same place.
2. Calculate the total time of flight of \(P\) and the total time of flight of \(Q\).
3. Calculate the range of the particles and hence calculate \(V _ { 2 }\) and \(\theta _ { 2 }\).