[In this question, the unit vectors $$\(\mathbf{i}\)$$ and $$\(\mathbf{j}\)$$ are in a vertical plane, with $$\(\mathbf{i}\)$$ being horizontal and $$\(\mathbf{j}\)$$ being vertically upwards.] At time $$\(t=0\)$$, a small ball is projected from a fixed point $$\(O\)$$ on horizontal ground. The ball is projected from $$\(O\)$$ with velocity $$\((p \mathbf{i}+q \mathbf{j}) \mathrm{m} \mathrm{s}^{-1}\)$$, where $$\(p\)$$ and $$\(q\)$$ are positive constants. The ball moves freely under gravity. At time $$\(t=3\)$$ seconds, the ball passes through the point $$\(A\)$$ with velocity $$\((8 \mathbf{i}-12 \mathbf{j}) \mathrm{m} \mathrm{s}^{-1}\)$$, as shown in Figure 4. (a) Find the speed of the ball at the instant it is projected from $$\(O\)$$. (5) For an interval of $$\(T\)$$ seconds the speed, $$\(v \mathrm{~m} \mathrm{~s}^{-1}\)$$, of the ball is such that $$\(v \leqslant 10\)$$ (b) Find the value of $$\(T\)$$. (4) At the point $$\(B\)$$ on the path of the ball, the direction of motion of the ball is perpendicular to the direction of motion of the ball at $$\(A\)$$. (c) Find the vertical height of $$\(B\)$$ above $$\(A\)$$. (4)

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