M5 June 2016 Q2
2. A particle \(P\) is moving in a plane. At time \(t\) seconds the position vector of \(P\) is \(\mathbf{r}\) metres and the velocity of \(P\) is \(\mathbf{v}\) m s\(^{-1}\). When \(t = \dfrac{\pi}{2}\), \(P\) is instantaneously at rest at the point with position vector \((\mathbf{i} - \mathbf{j})\) m.
Given that \(\mathbf{r}\) satisfies the differential equation
\[\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2} + 4\mathbf{r} = (3\sin t)\,\mathbf{i}\]find \(\mathbf{v}\) in terms of \(t\). (13)
| Scheme | Marks |
|---|---|
| \(\lambda^2 + 4 = 0\) | M1 |
| \(\mathbf{r} = \mathbf{A}\cos 2t + \mathbf{B}\sin 2t\) | A1 |
| PI: \(\ \mathbf{r} = \mathbf{c}\sin t + \mathbf{d}\cos t\) | B1 |
| \(\mathbf{r}^\prime = \mathbf{c}\cos t - \mathbf{d}\sin t\) | M1 |
| \(\mathbf{r}^{\prime\prime} = -\mathbf{c}\sin t - \mathbf{d}\cos t\) | A1 |
| \(-\mathbf{c}\sin t - \mathbf{d}\cos t + 4(\mathbf{c}\sin t + \mathbf{d}\cos t) \equiv 3\sin t\,\mathbf{i}\) | M1 |
| \(\mathbf{c} = \mathbf{i};\ \mathbf{d} = \mathbf{0}\) | M1 |
| GS is \(\mathbf{r} = \mathbf{A}\cos 2t + \mathbf{B}\sin 2t + \mathbf{i}\sin t\) | A1 |
| When \(t = \tfrac{\pi}{2},\ \mathbf{r} = \mathbf{i} - \mathbf{j} \Rightarrow \mathbf{A} = \mathbf{j}\) | M1 A1 |
| \(\mathbf{v} = -2\mathbf{A}\sin 2t + 2\mathbf{B}\cos 2t + \mathbf{i}\cos t\) | M1 |
| When \(t = \tfrac{\pi}{2},\ \mathbf{v} = \mathbf{0} \Rightarrow \mathbf{B} = \mathbf{0}\) | A1 |
| \(\mathbf{r} = \mathbf{i}\sin t + \mathbf{j}\cos 2t\) | |
| \(\mathbf{v} = \mathbf{i}\cos t - 2\mathbf{j}\sin 2t\) | A1 |
| (13 marks) |
Notes
First M1 for auxiliary equation
First A1 for correct CF (condone omission of \(\mathbf{r} =\) )
B1 for correct PI (they may realise \(\mathbf{d} = \mathbf{0}\) which is fine) (condone omission of \(\mathbf{r} =\) )
Second M1 for differentiating their PI
Second A1 for correct 2nd derivative
Third M1 for substituting into the DE
Fourth M1 for equating coeffs of \(\sin t\) and \(\cos t\) and finding \(\mathbf{c}\) and \(\mathbf{d}\)
Third A1 for correct GS with \(\mathbf{r} =\)
Fifth M1 for using conditions to find \(\mathbf{A}\)
Fourth A1 for \(\mathbf{A} = \mathbf{j}\)
Sixth M1 for differentiating \(\mathbf{r}\) to give \(\mathbf{v}\)
Fifth A1 for \(\mathbf{B} = \mathbf{0}\)
Sixth A1 for \(\mathbf{v} = \mathbf{i}\cos t - 2\mathbf{j}\sin 2t\)