M2 June 2009 Q2
2. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\), where
\[v = 8t - t^2.\](a) Find the maximum value of \(v\). (4)
(b) Find the time taken for \(P\) to return to \(O\). (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}v}{\mathrm{d}t} = 8 - 2t\) | M1 |
| \(8 - 2t = 0\) | M1 |
| Max \(v = 8 \times 4 - 4^2 = 16\) (m s\(^{-1}\)) | M1A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\displaystyle\int 8t - t^2\,\mathrm{d}t = 4t^2 - \dfrac{1}{3}t^3\ (+c)\) | M1A1 |
| (\(t = 0\), displacement \(= 0 \Rightarrow c = 0\)) | |
| \(4T^2 - \dfrac{1}{3}T^3 = 0\) | DM1 |
| \(T^2\left(4 - \dfrac{T}{3}\right) = 0 \Rightarrow T = 0, 12\) | DM1 |
| \(T = 12\) (seconds) | A1 |
| (5) | |
| (9 marks) |