A particle $$\(P\)$$ is moving along the $$\(x\)$$-axis. At time $$\(t\)$$ seconds, where $$\(t \geqslant 0, P\)$$ is $$\(x\)$$ metres from the origin $$\(O\)$$ and is moving with speed $$\(v \mathrm{~m} \mathrm{~s}^{-1}\)$$ The acceleration of $$\(P\)$$ has magnitude $$\(\frac{2}{(2 x+1)^{3}} \mathrm{~m} \mathrm{~s}^{-2}\)$$ and is directed towards $$\(O\)$$ When $$\(t=0, P\)$$ passes through $$\(O\)$$ in the positive $$\(x\)$$ direction with speed $$\(1 \mathrm{~m} \mathrm{~s}^{-1}\)$$ (a) Find $$\(v\)$$ in terms of $$\(x\)$$ (4) (b) Show that $$\(x=\frac{1}{2}(\sqrt{(4 t+1)}-1)\)$$ (4)

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