Displacement with an exponentially decaying sine factor, distance travelled and a geometric sequence of greatest distances

A particle \(P\) moves in a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is \(s\) cm, where \(s(t)=4\mathrm{e}^{-\frac{t}{4}}\sin\left(\frac{\pi t}{2}\right)\), for \(t\geq0\).

Part of the graph of \(s\) against \(t\) is shown in the diagram below.

1. Find
   1. the greatest displacement of \(P\) from \(O\);
   2. the greatest velocity reached by \(P\).[5]
2. Find
   1. the least value taken by \(s(t)\);
   2. the value of \(s\) when \(t=3.6\).[3]
3. Hence, find the total distance that \(P\) covers during the first \(3.6\) seconds.[3]

After leaving \(O\) at \(t=0\), the particle is next at \(O\) when \(t=T\).

4. Write down the value of \(T\).[1]

After this, \(P\) passes through \(O\) every \(T\) seconds. Let \(u_1,u_2,u_3,\ldots\) be the greatest distances of \(P\) from \(O\) during the time intervals \(0<t<T\), \(T<t<2T\), \(2T<t<3T,\ldots\) respectively. These distances form a geometric sequence.

5. 1. Find the common ratio, \(r\), of this sequence.
   2. Assuming that the motion carries on in the same pattern for ever, find the total distance that \(P\) covers.[5]