One end of a light inextensible string of length $$\(4 a\)$$ is attached to a fixed point $$\(A\)$$ on a horizontal ceiling. A particle, $$\(P\)$$, of mass $$\(m\)$$ is attached to the other end of the string. The particle is held in equilibrium at a vertical distance $$\(3 a\)$$ below the ceiling, with the string taut. The particle is then projected with speed $$\(\sqrt{7 a g}\)$$, in the direction perpendicular to the string, in the vertical plane containing $$\(A\)$$ and the string, as shown in Figure 2. In the subsequent motion the string remains taut. (a) Find the speed of $$\(P\)$$ at the instant before it hits the ceiling. (4) The point $$\(B\)$$ is the lowest point of the path of $$\(P\)$$. The first time $$\(P\)$$ passes through $$\(B\)$$ the tension in the string is $$\(T_{1}\)$$ and the second time $$\(P\)$$ passes through $$\(B\)$$ the tension in the string is $$\(T_{2}\)$$ Given that the coefficient of restitution between $$\(P\)$$ and the ceiling is $$\(\frac{1}{2}\)$$ (b) find the ratio $$\(T_{1}: T_{2}\)$$ in its simplest form. (7)

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