A particle \(P\) moves in a straight line and passes through a fixed point \(O\). At time \(t\) seconds, its displacement from \(O\), \(s\) metres, is given by

\(s=t+6t^2-t^3\) for \(0\leqslant t\leqslant3\).

\(s=12t-\frac13t^2-3\) for \(3\leqslant t\leqslant k\), where \(k\) is a constant.

It is given that, for \(3\leqslant t\leqslant k\), the velocity of \(P\) is positive and its acceleration is negative.

(a) The maximum velocity of \(P\) occurs when \(t=2\). On the axes below, sketch a velocity-time graph for the first \(k\) seconds of the motion of \(P\).

(b) The total distance travelled by \(P\) for \(0\leqslant t\leqslant k\) is 57 metres. Given that when \(t=3\) the distance and displacement of \(P\) from \(O\) are equal, find the value of \(k\).