5 A particle \(P\) of mass 2 kg moves along the \(x\)-axis.
At time \(t = 0 , P\) passes through the origin \(O\) with speed \(3 \mathrm {~ms} ^ { - 1 }\).
At time \(t\) seconds the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\), where \(t \geqslant 0 , x \geqslant 0\) and \(v \geqslant 0\).
While \(P\) is in motion the only force acting on \(P\) is a resistive force \(F\) of magnitude \(\left( v ^ { 2 } + 1 \right) \mathrm { N }\) acting in the negative \(x\)-direction.
- Find an expression for \(v\) in terms of \(x\).
- Determine the distance travelled by \(P\) while its speed drops from \(3 \mathrm {~ms} ^ { - 1 }\) to \(2 \mathrm {~ms} ^ { - 1 }\).
Particle \(Q\) is identical to particle \(P\). At a different time, \(Q\) is moving along the \(x\)-axis under the influence of a single constant resistive force of magnitude 1 N . When \(t ^ { \prime } = 0 , Q\) is at the origin and its speed is \(3 \mathrm {~ms} ^ { - 1 }\).
- By comparing the motion of \(P\) with the motion of \(Q\) explain why \(P\) must come to rest at some finite time when \(t < 6\) with \(x < 9\).
- Sketch the velocity-time graph for \(P\). You do not need to indicate any values on your sketch.
- Determine the maximum displacement of \(P\) from \(O\) during \(P\) 's motion.