M3 January 2006 Q2
2. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\) and its acceleration is \(2\sin\tfrac{1}{2}t\) m s\(^{-2}\), both measured in the direction of \(Ox\). Given that \(v = 4\) when \(t = 0\),
(a) find \(v\) in terms of \(t\), (4)
(b) calculate the distance travelled by \(P\) between the times \(t = 0\) and \(t = \dfrac{\pi}{2}\). (4)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}v}{\mathrm{d}t} = 2\sin\tfrac{1}{2}t \Rightarrow v = A - 4\cos\tfrac{1}{2}t\) | M1 A1 |
| \(v = 4,\ t = 0 \Rightarrow 4 = A - 4 \Rightarrow A = 8\) | M1 |
| \(v = 8 - 4\cos\tfrac{1}{2}t\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\displaystyle\int_{\ldots}^{\ldots}\left(8 - 4\cos\tfrac{1}{2}t\right)\mathrm{d}t = 8t - 8\sin\tfrac{1}{2}t\) ft constants | M1 A1ft |
| \(\left[\ldots\right]_0^{\pi/2} = 4\left(\pi - \sqrt{2}\right)\) awrt 6.9 | M1 A1 |
| (4) | |
| (8 marks) |