A particle, $$\(P\)$$, is projected vertically upwards with speed $$\(U\)$$ from a fixed point $$\(O\)$$. At the instant when $$\(P\)$$ reaches its greatest height $$\(H\)$$ above $$\(O\)$$, a second particle, $$\(Q\)$$, is projected with speed $$\(\frac{1}{2} U\)$$ vertically upwards from $$\(O\)$$. (a) Find $$\(H\)$$ in terms of $$\(U\)$$ and $$\(g\)$$. (2) (b) Find, in terms of $$\(U\)$$ and $$\(g\)$$, the time between the instant when $$\(Q\)$$ is projected and the instant when the two particles collide. (6) (c) Find where the two particles collide. (3)

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