A light elastic string has a natural length of $3\,\text{m}$ and a modulus of elasticity $45\,\text{N}$. A particle $P$ with weight $6\,\text{N}$ is fixed to the midpoint of the string. The two ends of the string are fastened to fixed points $A$ and $B$ on the same vertical line, with $A$ above $B$ and $AB = 4\,\text{m}$. The particle $P$ is let go from rest at a position $1.5\,\text{m}$ vertically beneath $A$.
(i)[4]
Calculate the distance $P$ travels after being released before it first comes to instantaneous rest at a point vertically above $B$. (You may assume that, at that instant, the segment of string connecting $P$ to $B$ is slack.)
(ii)[5]
Show that the greatest speed of $P$ is reached when it is $2.1\,\text{m}$ below $A$, and calculate this greatest speed.
(iii)[3]
Calculate the largest magnitude of the acceleration of $P$.
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