June 2025 Paper 2 Q19
19 The displacement, \(\mathbf{s}\) metres, of a particle \(P\), at time \(t\) seconds, is given by
\[\mathbf{s} = (2t^3)\,\mathbf{i} + (2t^2 + qt)\,\mathbf{j}\](a) Find an expression for the velocity of particle \(P\) after \(t\) seconds. [2 marks]
(b) The acceleration, \(\mathbf{a}\ \text{m s}^{-2}\), of a particle \(Q\), at time \(t\) seconds, is given by\[\mathbf{a} = 6t\,\mathbf{i} + 7\mathbf{j}\]
Particle \(Q\) has an initial velocity of \(4\mathbf{j}\ \text{m s}^{-1}\)
Particles \(P\) and \(Q\) are moving parallel to each other when \(t = 2\)
Find the value of \(q\)
[6 marks]| Scheme | Marks | AO |
|---|---|---|
| Uses \(\mathbf{v} = \dfrac{\mathrm{d}\mathbf{s}}{\mathrm{d}t}\) with one component correct | M1 | 3.4 |
| Obtains \(6t^2\mathbf{i} + (4t + q)\mathbf{j}\) OE | A1 | 1.1b |
| (2) |
Typical solution
\[\mathbf{v} = \frac{\mathrm{d}\mathbf{s}}{\mathrm{d}t} = 6t^2\mathbf{i} + (4t + q)\mathbf{j}\]| Scheme | Marks | AO |
|---|---|---|
| Uses \(\mathbf{v} = \displaystyle\int \mathbf{a}\,\mathrm{d}t\) with one component correct Ignore c | M1 | 3.4 |
| Substitutes \(t = 0\) into a velocity vector and equates to \(4\mathbf{j}\) to obtain a value or values for their constant of integration. PI by \(3t^2\mathbf{i} + (7t + 4)\mathbf{j}\) | M1 | 3.4 |
| Obtains \(3t^2\mathbf{i} + (7t + 4)\mathbf{j}\) | A1 | 1.1b |
| Finds their \(\mathbf{v}_p\) and their \(\mathbf{v}_q\) when \(t = 2\). Must come from use of calculus Condone missing brackets for \(8 + q\) | B1F | 1.1b |
| Uses \(\mathbf{v}_p = k\mathbf{v}_q\) Where \(k \neq 1\) Or Compares the ratios of the components of their \(\mathbf{v}_p\) and their \(\mathbf{v}_q\) Must come from use of calculus | M1 | 3.3 |
| Obtains 28 | A1 | 1.1b |
| (6) | ||
| (8 marks) |
Typical solution
\[\mathbf{a} = \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t},\ \mathbf{v} = \int 6t\,\mathbf{i} + 7\mathbf{j}\,\mathrm{d}t\]\[\mathbf{v} = 3t^2\mathbf{i} + 7t\mathbf{j} + c_1\mathbf{i} + c_2\mathbf{j}\]When \(t = 0,\ \mathbf{v} = 4\mathbf{j},\ c_2 = 4\)
\[\mathbf{v} = 3t^2\mathbf{i} + (7t + 4)\mathbf{j}\]When \(t = 2\),
\[\mathbf{v}_p = 24\mathbf{i} + (8 + q)\mathbf{j}\]\[\mathbf{v}_q = 12\mathbf{i} + 18\mathbf{j}\]\[\therefore \mathbf{v}_p = 2\mathbf{v}_q\]\[36 = 8 + q\]\[q = 28\]