M2 January 2011 Q3
3. A particle moves along the \(x\)-axis. At time \(t = 0\) the particle passes through the origin with speed 8 m s\(^{-1}\) in the positive \(x\)-direction. The acceleration of the particle at time \(t\) seconds, \(t \geqslant 0\), is \((4t^3 - 12t)\) m s\(^{-2}\) in the positive \(x\)-direction.
Find
(a) the velocity of the particle at time \(t\) seconds, (3)
(b) the displacement of the particle from the origin at time \(t\) seconds, (2)
(c) the values of \(t\) at which the particle is instantaneously at rest. (3)
| Scheme | Marks |
|---|---|
| \(a = 4t^3 - 12t\) | |
| Convincing attempt to integrate | M1 |
| \(v = t^4 - 6t^2\ (+c)\) | A1 |
| Use initial condition to get \(\ v = t^4 - 6t^2 + 8\left(\text{ms}^{-1}\right)\). | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Convincing attempt to integrate | M1 |
| \(s = \dfrac{t^5}{5} - 2t^3 + 8t\ (+0)\) Integral of their \(v\) | A1ft |
| (2) |
| Scheme | Marks |
|---|---|
| Set their \(\ v = 0\) | M1 |
| Solve a quadratic in \(\ t^2\) | DM1 |
| \((t^2 - 2)(t^2 - 4) = 0 \Rightarrow\) at rest when \(\ t = \sqrt{2},\ t = 2\) | A1 |
| (3) | |
| (8 marks) |