M3 January 2010 Q1
1. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis. At time \(t\) seconds, \(P\) is moving under the action of a single force of magnitude \(\left[4 + \cos(\pi t)\right]\) N, directed away from the origin. When \(t = 1\), the particle \(P\) is moving away from the origin with speed 6 m s\(^{-1}\).
Find the speed of \(P\) when \(t = 1.5\), giving your answer to 3 significant figures. (7)
| Scheme | Marks |
|---|---|
| \(0.5a = 4 + \cos(\pi t)\) | B1 |
| Integrating \(\quad 0.5v = 4t + \dfrac{\sin(\pi t)}{\pi}\ \ (+C)\) | M1 A1 |
| Using boundary values \(3 = 4 + C \Rightarrow C = -1\) | M1 A1 |
| When \(t = 1.5\) \(0.5v = 6 - \dfrac{1}{\pi} - 1\) | M1 |
| \(v \approx 9.36\ \left(\text{m s}^{-1}\right)\) cao | A1 |
| (7) | |
| (7 marks) |