At time $$\(t=0\)$$ a particle $$\(P\)$$ is projected from a fixed point $$\(A\)$$ on horizontal ground. The particle is projected with speed $$\(25 \mathrm{~m} \mathrm{~s}^{-1}\)$$ at an angle $$\(\alpha\)$$ to the ground. The particle moves freely under gravity. At time $$\(t=3\)$$ seconds, $$\(P\)$$ is passing through the point $$\(B\)$$ with speed $$\(15 \mathrm{~m} \mathrm{~s}^{-1}\)$$ and is moving downwards at an angle $$\(\beta\)$$ to the horizontal, as shown in Figure 5. (a) By considering energy, find the height of $$\(B\)$$ above the ground. (3) (b) Find the size of angle $$\(\alpha\)$$. (3) (c) Find the size of angle $$\(\beta\)$$. (3) (d) Find the least speed of $$\(P\)$$ as $$\(P\)$$ travels from $$\(A\)$$ to $$\(B\)$$. (2) As $$\(P\)$$ travels from $$\(A\)$$ to $$\(B\)$$, the speed, $$\(v \mathrm{~m} \mathrm{~s}^{-1}\)$$, of $$\(P\)$$ is such that $$\(v \leqslant 15\)$$ for an interval of $$\(T\)$$ seconds. (e) Find the value of $$\(T\)$$. (3)

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