M2 June 2010 Q1
1. A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds, \(t \geqslant 0\), is \((3t + 5)\) m s\(^{-2}\) in the positive \(x\)-direction. When \(t = 0\), the velocity of \(P\) is 2 m s\(^{-1}\) in the positive \(x\)-direction. When \(t = T\), the velocity of \(P\) is 6 m s\(^{-1}\) in the positive \(x\)-direction. Find the value of \(T\). (6)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}v}{\mathrm{d}t} = 3t + 5\) | |
| \(v = \displaystyle\int (3t + 5)\,\mathrm{d}t\) | M1* |
| \(v = \frac{3}{2}t^2 + 5t\ \ (+\,c)\) | A1 |
| \(t = 0\ \ v = 2\ \ \Rightarrow\ \ c = 2\) | B1 |
| \(v = \frac{3}{2}t^2 + 5t + 2\) | |
| \(t = T\qquad 6 = \frac{3}{2}T^2 + 5T + 2\) | DM1* |
| \(12 = 3T^2 + 10T + 4\) | |
| \(3T^2 + 10T - 8 = 0\) | |
| \((3T - 2)(T + 4) = 0\) | M1 |
| \(T = \frac{2}{3}\quad (T = -4)\) | |
| \(\therefore T = \frac{2}{3}\quad\) (or 0.67) | A1 |
| (6 marks) |