M3 January 2011 Q1
1. A particle \(P\) moves on the positive \(x\)-axis. When the distance of \(P\) from the origin \(O\) is \(x\) metres, the acceleration of \(P\) is \((7 - 2x)\) m s\(^{-2}\), measured in the positive \(x\)-direction. When \(t = 0\), \(P\) is at \(O\) and is moving in the positive \(x\)-direction with speed 6 m s\(^{-1}\). Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest. (6)
| Scheme | Marks |
|---|---|
| \(v\dfrac{\mathrm{d}v}{\mathrm{d}x} = 7 - 2x\) | M1 |
| \(\tfrac{1}{2}v^2 = 7x - x^2 \quad (+c)\) | M1A1 |
| \(x = 0 \quad v = 6 \quad\Rightarrow\quad c = 18\) | A1 |
| \(v = 0 \quad x^2 - 7x - 18 = 0\) \((x + 2)(x - 9) = 0\) | M1 |
| \(\therefore x = 9\) | A1 |
| (6 marks) |