M2 January 2010 Q1
1. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\) in the positive \(x\)-direction, where \(v = 3t^2 - 4t + 3\). When \(t = 0\), \(P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) is moving with minimum velocity. (8)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}v}{\mathrm{d}t} = 6t - 4\) | M1 A1 |
| \(6t - 4 = 0 \Rightarrow t = \frac{2}{3}\) | M1 A1 |
| \(s = \displaystyle\int 3t^2 - 4t + 3\ \mathrm{d}t = t^3 - 2t^2 + 3t\ (+\,c)\) | M1 A1 |
| \(t = \frac{2}{3} \Rightarrow s = -\frac{16}{27} + 2\) so distance is \(\frac{38}{27}\) m | M1 A1 |
| (8 marks) |