Problem

A particle $Q$ is initially positioned at a distance $d$ vertically above a particle $P$. Particle $P$ is projected with speed $U$ at an angle $\alpha$ above the horizontal. At the same time, $Q$ is projected at an angle $\beta$ below the horizontal. Both particles move freely under gravity. The particles collide at time $T$ after the projections.

(a) Show that

$$d=UT(\sin\alpha+\cos\alpha\tan\beta).$$

The particles collide when $P$ is at its maximum height.

(b) Given that $\alpha=30^\circ$ and $\beta=60^\circ$, find $d$ in terms of $U$ and $g$.