M3 January 2009 Q1
1. A particle \(P\) of mass 3 kg is moving in a straight line. At time \(t\) seconds, \(0 \leqslant t \leqslant 4\), the only force acting on \(P\) is a resistance to motion of magnitude \(\left(9 + \dfrac{15}{(t + 1)^2}\right)\) N. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\). When \(t = 4\), \(v = 0\).
Find the value of \(v\) when \(t = 0\). (7)
| Scheme | Marks |
|---|---|
| N2L \(3a = -\left(9 + \dfrac{15}{(t + 1)^2}\right)\) | B1 |
| \(3v = -9t + \dfrac{15}{t + 1}\ (+A)\) | M1 A1ft |
| \(v = 0,\ t = 4 \Rightarrow 0 = -36 + 3 + A \Rightarrow A = 33\) | M1 A1 |
| \(v = -3t + \dfrac{5}{t + 1} + 11\) | |
| \(t = 0 \Rightarrow v = 16\) | M1 A1 |
| (7) | |
| (7 marks) |