June 2022 Paper 2 Q17
17 A particle is moving such that its position vector, \(\mathbf{r}\) metres, at time \(t\) seconds, is given by
\[\mathbf{r} = \mathrm{e}^t\cos t\,\mathbf{i} + \mathrm{e}^t\sin t\,\mathbf{j}\]Show that the magnitude of the acceleration of the particle, \(a\) m s−2, is given by
\[a = 2\mathrm{e}^t\]Fully justify your answer. [7 marks]
| Scheme | Marks | AO |
|---|---|---|
| Differentiates with evidence of correct use of product rule. Condone sign errors | M1 | 3.4 |
| Finds expression for \(\mathbf{v}\) or \(\dfrac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\) with either \(\mathbf{i}\) or \(\mathbf{j}\) component fully correct | M1 | 1.1a |
| Finds fully correct expression for \(\mathbf{v}\) or \(\dfrac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\) \((\mathrm{e}^t\cos t - \mathrm{e}^t\sin t)\mathbf{i} + (\mathrm{e}^t\sin t + \mathrm{e}^t\cos t)\mathbf{j}\) Condone poor use of brackets provided fully correct acceleration seen | A1 | 1.1b |
| Differentiates their \(\mathbf{v}\) or \(\dfrac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\) with evidence of correct use of product rule to find an expression for \(\mathbf{a}\) with at least one component correct. Condone sign errors | M1 | 3.4 |
| Finds correct expression for \(\mathbf{a}\) May be unsimplified | A1 | 1.1b |
| Obtains an expression for the magnitude of their \(\mathbf{a}\) provided their \(\mathbf{a}\) has non-zero \(\mathbf{i}\) and \(\mathbf{j}\) components | M1 | 1.1a |
| Completes reasoned argument from a correct \(\mathbf{a}\) to show given result. Must see a factor of \((\sin^2 t + \cos^2 t)\) eg \(\sqrt{4\mathrm{e}^{2t}(\sin^2 t + \cos^2 t)}\) AG | R1 | 2.1 |
| (7 marks) |