- In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
A particle \(P\) moves along a straight line. Initially \(P\) is at rest at the point \(O\) on the line.
At time \(t\) seconds, where \(t \geqslant 0\)
- the displacement of \(P\) from \(O\) is \(x\) metres
- the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction
- the acceleration of \(P\) is \(\frac { 96 } { ( 3 t + 5 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction
- Show that, at time \(t\) seconds, \(v = p - \frac { q } { ( 3 t + 5 ) ^ { 2 } }\), where \(p\) and \(q\) are constants to be determined.
- Find the limiting value of \(v\) as \(t\) increases.
- Find the value of \(x\) when \(t = 2\)