In this question, the units are metres and seconds. A particle \(P\) is travelling in a straight line through a fixed point \(O\).

At time \(t\) its acceleration, \(a\), is given by \(a=(2t-3)^2\), where \(t\geqslant0\). When \(t=3\), \(P\) has a velocity of \(6\).

(a)(i) Find an expression for the velocity, \(v\), of \(P\) at time \(t\).

(ii) Find the time when \(P\) is at rest.

When \(t=\frac52\), the displacement of \(P\) from \(O\) is \(4\).

(iii) Find the displacement of \(P\) from \(O\) when \(t=3\).

(b) Use calculus to find the approximate change in \(v\) when \(t\) increases from \(\frac52\) by the small amount \(0.02\).