June 2022 Paper 3 Mechanics Q1
1. [In this question, position vectors are given relative to a fixed origin.]
At time \(t\) seconds, where \(t > 0\), a particle \(P\) has velocity \(\mathbf{v}\ \text{m s}^{-1}\) where
\[\mathbf{v} = 3t^2\mathbf{i} - 6t^{\frac{1}{2}}\mathbf{j}\]At time \(t = 4\) seconds, the position vector of \(P\) is \((\mathbf{i} - 4\mathbf{j})\) m.
| Scheme | Marks | AO |
|---|---|---|
| Put \(t = 2\) in \(\mathbf{v}\) and use Pythagoras: \(\sqrt{12^2 + (-6\sqrt{2})^2}\) | M1 | 3.1a |
| \(\sqrt{216},\ 6\sqrt{6}\) or 15 or better \((\text{m s}^{-1})\) | A1 | 1.1b |
| (2) |
Notes
Accept column vectors throughout apart from the answer to (b).
M1: Need square root but -ve sign not required. Allow \(\mathbf{i}\)’s and/or \(\mathbf{j}\)’s to go missing from their \(\mathbf{v}\) at \(t = 2\), provided they have applied Pythagoras correctly.
A1: cao
N.B. Correct answer with no working can score 2 marks.
| Scheme | Marks | AO |
|---|---|---|
| Differentiate \(\mathbf{v}\) wrt \(t\) to obtain \(\mathbf{a}\) | M1 | 3.4 |
| \(6t\mathbf{i} - 3t^{-\frac{1}{2}}\mathbf{j}\) oe \((\text{m s}^{-2})\) isw | A1 | 1.1b |
| (2) |
Notes
Accept column vectors throughout apart from the answer to (b).
M1: Both powers decreasing by 1. Allow a column vector.
M0 if \(\mathbf{i}\) or \(\mathbf{j}\) is missing but allow recovery in (b).
A1: cao. Do not accept a column vector.
| Scheme | Marks | AO |
|---|---|---|
| Integrate \(\mathbf{v}\) wrt \(t\) to obtain \(\mathbf{r}\) | M1 | 3.4 |
| \(\mathbf{r} = t^3\mathbf{i} - 4t^{\frac{3}{2}}\mathbf{j}\ (+\mathbf{C})\) | A1 | 1.1b |
| \((\mathbf{i} - 4\mathbf{j}) = 4^3\mathbf{i} - 4 \times 4^{\frac{3}{2}}\mathbf{j} + \mathbf{C}\) | M1 | 3.1a |
| \((-62\mathbf{i} + 24\mathbf{j})\) (m) isw e.g. if they go on to find the distance. | A1 | 1.1b |
| (4) | ||
| (8 marks) |
Notes
Accept column vectors throughout apart from the answer to (b).
M1: Both powers increasing by 1
M0 if \(\mathbf{i}\) or \(\mathbf{j}\) is missing but allow recovery.
A1: (\(\mathbf{r} =\)) not required
M1: Putting \(\mathbf{r} = (\mathbf{i} - 4\mathbf{j})\) and \(t = 4\) into their displacement vector expression which must have \(\mathbf{C}\) (allow \(C\)) to give an equation in \(\mathbf{C}\) only, seen or implied.
Must have attempted to integrate \(\mathbf{v}\) for this mark to be available.
N.B. \(\mathbf{C}\) does not need to be found and this is a method mark, so allow slips.
A1: cao