Two particles $P$ and $Q$ are projected vertically upwards from horizontal ground at the same instant.
The speeds of projection of $P$ and $Q$ are $12{\text{ }}m{\text{ }}{s^{ - 1}}$ and $7{\text{ }}m{\text{ }}{s^{ - 1}}$ respectively and the heights of $P$ and $Q$ above the ground, $t$ seconds after projection, are ${h_P}m$ and ${h_Q}m$ respectively. Each particle comes to rest on returning to the ground.
a) Find the set of values of $t$ for which the particles are travelling in opposite directions.
b) At a certain instant, $P$ and $Q$ are above the ground and $3{h_P} = 8{h_Q}$. Find the velocities of $P$ and $Q$ at this instant.
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