7 A particle \(P\) travels in a straight line, starting at rest from a point \(O\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is denoted by \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where
$$\begin{array} { l l }
a = 0.3 t ^ { \frac { 1 } { 2 } } & \text { for } 0 \leqslant t \leqslant 4 ,
a = - k t ^ { - \frac { 3 } { 2 } } & \text { for } 4 < t \leqslant T ,
\end{array}$$
where \(k\) and \(T\) are constants.
- Find the velocity of \(P\) at \(t = 4\).
- It is given that there is no change in the velocity of \(P\) at \(t = 4\) and that the velocity of \(P\) at \(t = 16\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Show that \(k = 2.6\) and find an expression, in terms of \(t\), for the velocity of \(P\) for \(4 \leqslant t \leqslant T\).
- Given that \(P\) comes to instantaneous rest at \(t = T\), find the exact value of \(T\).
- Find the total distance travelled between \(t = 0\) and \(t = T\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.