A particle \(P\) is moving along the positive \(x\)-axis. At time \(t = 0 , P\) is at the origin \(O\). At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and has velocity \(v = 2 \mathrm { e } ^ { - x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing.
Find the acceleration of \(P\) in terms of \(x\).
Find \(x\) in terms of \(t\).
A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\). The period of the motion is \(\frac { \pi } { 2 }\) seconds. At time \(t\) seconds the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0 , P\) is at \(O\) and \(v = 6\). Find
the greatest distance of \(P\) from \(O\) during the motion,
the greatest magnitude of the acceleration of \(P\) during the motion,
the smallest positive value of \(t\) for which \(P\) is 1 m from \(O\).