| Exam Board | AQA |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2008 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Relative velocity: find resultant velocity (magnitude and/or direction) |
| Difficulty | Moderate -0.8 This is a straightforward application of Pythagoras' theorem to perpendicular velocity vectors (8² + U² = 10²) followed by basic trigonometry for the bearing. It requires only routine vector addition concepts with no problem-solving insight, making it easier than average for A-level. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(U = \sqrt{10^2 - 8^2} = 6\) | M1, A1 (Total: 2) | Expression/equation for \(U\) based on a right angled triangle. Correct \(U\). Note \(10^2 + 8^2\) gives M1A0. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos \theta = \frac{8}{10}\) → \(\theta = 037°\) | M1, A1 (Total: 2) | Use of trigonometry to find angle. Allow \(\tan \theta = \frac{8}{6}\) or \(\frac{6}{8}\) or \(\sin/\cos \theta = \frac{8}{10}\) or \(\frac{6}{10}\). Correct angle. Accept 36.9° etc. Note 143° gives M1A0. |
### Part (a)
$U = \sqrt{10^2 - 8^2} = 6$ | M1, A1 (Total: 2) | Expression/equation for $U$ based on a right angled triangle. Correct $U$. Note $10^2 + 8^2$ gives M1A0.
### Part (b)
$\cos \theta = \frac{8}{10}$ → $\theta = 037°$ | M1, A1 (Total: 2) | Use of trigonometry to find angle. Allow $\tan \theta = \frac{8}{6}$ or $\frac{6}{8}$ or $\sin/\cos \theta = \frac{8}{10}$ or $\frac{6}{10}$. Correct angle. Accept 36.9° etc. Note 143° gives M1A0.
---2 The velocity of a ship, relative to the water in which it is moving, is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ due north. The water is moving due east with a speed of $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resultant velocity of the ship has magnitude $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Find $U$.
\item Find the direction of the resultant velocity of the ship. Give your answer as a bearing to the nearest degree.
\end{enumerate}
\hfill \mbox{\textit{AQA M1 2008 Q2 [4]}}