M3 January 2013 Q1
1. A particle \(P\) is moving along the positive \(x\)-axis. When the displacement of \(P\) from the origin is \(x\) metres, the velocity of \(P\) is \(v\) m s\(^{-1}\) and the acceleration of \(P\) is \(9x\) m s\(^{-2}\).
When \(x = 2\), \(v = 6\)
Show that \(v^2 = 9x^2\). (4)
| Scheme | Marks |
|---|---|
| \(v\dfrac{\mathrm{d}v}{\mathrm{d}x} = 9x\) | M1 |
| \(\dfrac{1}{2}v^2 = 9x\ \ \ (+c)\) | A1 |
| \(v^2 = 9x^2 + c\) | M1dep |
| \(x = 2\ \ v = 6\ \ \ \ 36 = 9 \times 4 + c \Rightarrow c = 0\) | |
| \(v^2 = 9x^2\) | A1 |