A light elastic string has natural length $$\(2 a\)$$ and modulus of elasticity $$\(2 m g\)$$. One end of the elastic string is attached to a fixed point $$\(O\)$$. A particle $$\(P\)$$ of mass $$\(\frac{1}{2} m\)$$ is attached to the other end of the elastic string. The point $$\(A\)$$ is vertically below $$\(O\)$$ with $$\(O A=4 a\)$$. Particle $$\(P\)$$ is held at $$\(A\)$$ and released from rest. The speed of $$\(P\)$$ at the instant when it has moved a distance $$\(a\)$$ upwards is $$\(\sqrt{3 a g}\)$$ Air resistance to the motion of $$\(P\)$$ is modelled as having magnitude $$\(k m g\)$$, where $$\(k\)$$ is a constant. Using the model and the work-energy principle, (a) show that $$\(k=\frac{1}{4}\)$$ (7) Particle $$\(P\)$$ is now held at $$\(O\)$$ and released from rest. As $$\(P\)$$ moves downwards, it reaches its maximum speed as it passes through the point $$\(B\)$$. (b) Find the distance $$\(O B\)$$. (4)

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