June 2018 Paper 3 Q6
6. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves in the \(x\)-\(y\) plane in such a way that its velocity \(\mathbf{v}\ \text{m s}^{-1}\) is given by
\[\mathbf{v} = t^{-\frac{1}{2}}\mathbf{i} - 4t\mathbf{j}\]When \(t = 1\), \(P\) is at the point \(A\) and when \(t = 4\), \(P\) is at the point \(B\).
Find the exact distance \(AB\). (6)
| Scheme | Marks | AO |
|---|---|---|
| Integrate \(\mathbf{v}\) w.r.t. time | M1 | 1.1a |
| \(\mathbf{r} = 2t^{\frac{1}{2}}\mathbf{i} - 2t^2\mathbf{j}\ \ (+\ \mathbf{C})\) | A1 | 1.1b |
| Substitute \(t = 4\) and \(t = 1\) into their \(\mathbf{r}\) | M1 | 1.1b |
| \(t = 4, \mathbf{r} = 4\mathbf{i} - 32\mathbf{j} (+\ \mathbf{C});\ t = 1, \mathbf{r} = 2\mathbf{i} - 2\mathbf{j} (+\ \mathbf{C})\) or \((4, -32);\ (2, -2)\) | A1 | 1.1b |
| \(\sqrt{2^2 + (-30)^2}\) | M1 | 1.1b |
| \(\sqrt{904} = 2\sqrt{226}\) | A1 | 1.1b |
| (6) | ||
| (6 marks) |
Notes
Allow column vectors throughout
M1: At least one power increasing by 1.
A1: Any correct (unsimplified) expression
M1: Must have attempted to integrate \(\mathbf{v}\). Substitute \(t = 4\) and \(t = 1\) into their \(\mathbf{r}\) to produce 2 vectors (or 2 points if just working with coordinates).
A1: \(4\mathbf{i} - 32\mathbf{j} (+\ \mathbf{C})\) and \(2\mathbf{i} - 2\mathbf{j} (+\ \mathbf{C})\) or \((4, -32)\) and \((2, -2)\). These can be seen or implied.
M1: Attempt at distance of form \(\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\) for their points. Must have 2 non zero terms.
A1: \(\sqrt{904} = 2\sqrt{226}\) or any equivalent surd (exact answer needed)