June 2025 Paper 3 Mechanics Q4
4. [In this question, position vectors are given relative to a fixed origin \(O\).]
At time \(t\) seconds, where \(t \gt 0\), the position vector of a particle \(P\) is \(\mathbf{r}\) metres where
\[\mathbf{r} = 4t^{\frac{3}{2}}\mathbf{i} - t^2\mathbf{j}\]At \(t = T\), the acceleration of \(P\) is in a direction that is perpendicular to the line with equation \(y = \dfrac{1}{3}x\)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{r} = (32\mathbf{i} - 16\mathbf{j})\) | B1 | 1.1b |
| (1) |
Notes
N.B. Accept column vectors throughout apart from in the answer to (c).
B1: cao.
N.B. Must be using correct vector notation, including brackets if using column vectors.
| Scheme | Marks | AO |
|---|---|---|
| Use Pythagoras: \(\sqrt{32^2 + 16^2}\) oe for their \(\mathbf{r}\) | M1 | 3.1a |
| \(16\sqrt{5}\) (m) | A1 | 1.1b |
| (2) |
Notes
N.B. Accept column vectors throughout apart from in the answer to (c).
M1: For an unsimplified expression, using their \(\mathbf{r}\), with the square root
A1: Accept any surd equivalent isw
N.B. Must come from \(\mathbf{r} = (32\mathbf{i} - 16\mathbf{j})\)
| Scheme | Marks | AO |
|---|---|---|
| Differentiate \(\mathbf{r}\) wrt \(t\) to obtain \(\mathbf{v}\) | M1 | 3.4 |
| \(6t^{\frac{1}{2}}\mathbf{i} - 2t\mathbf{j}\) oe \((\text{m s}^{-1})\) | A1 | 1.1b |
| (2) |
Notes
N.B. Accept column vectors throughout apart from in the answer to (c).
M1: Both powers of \(t\) decreasing by 1 (but not just division by \(t\))
N.B. M0 if \(\mathbf{i}\) and/or \(\mathbf{j}\) are missing and never reappear.
A1: Must be in terms of \(t\), \(\mathbf{i}\) and \(\mathbf{j}\)
| Scheme | Marks | AO |
|---|---|---|
| Differentiate their \(\mathbf{v}\) wrt \(t\) to obtain \(\mathbf{a}\) | M1 | 3.4 |
| \(3t^{-\frac{1}{2}}\mathbf{i} - 2\mathbf{j}\) | A1 | 1.1b |
| \(\dfrac{3T^{-\frac{1}{2}}}{-2} = -\dfrac{1}{3}\) must see \(\dfrac{\mathbf{i}\text{ component of their }\mathbf{a}}{\mathbf{j}\text{ component of their }\mathbf{a}} = -\dfrac{1}{3}\) oe | M1 | 2.1 |
| \((T =)\ \dfrac{81}{4}\) oe | A1 | 1.1b |
| (4) | ||
| (9 marks) |
Notes
N.B. Accept column vectors throughout apart from in the answer to (c).
M1: Both powers of \(t\) decreasing by 1 (but not just division by \(t\))
If no i’s and/or j’s, can score M1 if
EITHER they have 2 separate components, provided they are clearly treated as such in the subsequent working
OR the i’s and j’s reappear
otherwise M0.
A1: Correct vector or
2 correct separate components, provided they are clearly treated as such in the subsequent working
M1: Complete method to form a correct equation in \(T\) (\(t\)) only, for their a, using the 2 components from their a
A1: Accept 20.3 or 20.25 or any equivalent fraction (allow \(t\) instead of \(T\))