M2 January 2009 Q4
4. A particle \(P\) moves along the \(x\)-axis in a straight line so that, at time \(t\) seconds, the velocity of \(P\) is \(v\) m s\(^{-1}\), where
\[v = \begin{cases} 10t - 2t^2, & 0 \leqslant t \leqslant 6,\\[4pt] \dfrac{-432}{t^2}, & t > 6.\end{cases}\]At \(t = 0\), \(P\) is at the origin \(O\). Find the displacement of \(P\) from \(O\) when
(a) \(t = 6\), (3)
(b) \(t = 10\). (5)
| Scheme | Marks |
|---|---|
| \(v = 10t - 2t^2,\ \ s = \displaystyle\int v\,\mathrm{d}t\) | M1 |
| \(= 5t^2 - \dfrac{2t^3}{3}\ (+C)\) | A1 |
| \(t = 6 \Rightarrow s = 180 - 144 = 36\) (m) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(s = \displaystyle\int v\,\mathrm{d}t = \dfrac{-432t^{-1}}{-1}\ (+K) = \dfrac{432}{t}\ (+K)\) | B1 |
| \(t = 6\), \(s =\) “36” \(\Rightarrow 36 = \dfrac{432}{6} + K\) | M1* |
| \(\Rightarrow K = -36\) | A1 |
| At \(t = 10\), \(s = \dfrac{432}{10} - 36 = 7.2\) (m) | d*M1 A1 |
| (5) | |
| (8 marks) |