June 2024 Paper 3 Q8
8 A particle \(P\) is moving with constant acceleration \((-5\mathbf{i} + 2\mathbf{j})\,\mathrm{m\,s^{-2}}\). At time \(t = 0\) seconds, \(P\) is at the origin and has velocity \((\mathbf{i} + 3\mathbf{j})\,\mathrm{m\,s^{-1}}\).
| Scheme | Marks | AO |
|---|---|---|
| \([\mathbf{s} =]\ 2(\mathbf{i} + 3\mathbf{j}) + 0.5 \times 2^2 \times (-5\mathbf{i} + 2\mathbf{j})\) or \(2\begin{pmatrix}1\\3\end{pmatrix} + 0.5 \times 2^2 \times \begin{pmatrix}-5\\2\end{pmatrix}\) | M1 | 3.3 |
| \([\mathbf{s} =]\ -8\mathbf{i} + 10\mathbf{j}\) (m) | A1 | 1.1 |
| [2] |
Notes
M1: Apply \(\mathbf{s} = \mathbf{u}t + 0.5\mathbf{a}t^2\) correctly with correct values of \(\mathbf{u}\), \(\mathbf{a}\) and \(t\) – if using integration then for this mark we must see the correct expression \(\begin{pmatrix}1\\3\end{pmatrix}t + \frac{1}{2} \times \begin{pmatrix}-5\\2\end{pmatrix}t^2\) with \(t = 2\) subst.
A1: or \(\begin{pmatrix}-8\\10\end{pmatrix}\)
ISW if correct vector converted to scalar
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{v} = (\mathbf{i} + 3\mathbf{j}) + 2(-5\mathbf{i} + 2\mathbf{j})\) | M1* | 3.3 |
| \(\mathbf{v} = -9\mathbf{i} + 7\mathbf{j}\) | A1 | 1.1 |
| \(|\mathbf{v}| = \sqrt{(-9)^2 + 7^2}\) | M1dep* | 3.4 |
| \(|\mathbf{v}| = 11.4\ (\mathrm{m\,s^{-1}})\) | A1 | 1.1 |
| [4] |
Notes
M1*: Apply \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\) with correct values of \(\mathbf{u}\), \(\mathbf{a}\) and \(t\) (or other complete method to find \(\mathbf{v}\))
Allow from integration but must have correct expression for \(\mathbf{v}\) with \(t = 2\) substituted
A1: or as a column vector (possibly implied by correct magnitude)
M1dep*: Correct method for the speed of \(P\) at time \(t = 2\) – condone \(\sqrt{-9^2 + 7^2} = \sqrt{\pm 81 + 49}\)
A1: Allow \(\sqrt{130}\) or awrt 11.4 www – must follow from correct \(\mathbf{v} = -9\mathbf{i} + 7\mathbf{j}\) (so M1 A0 M1 A1 is not possible)
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