| Exam Board | AQA |
|---|---|
| Module | M1 (Mechanics 1) |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | River crossing: perpendicular heading or minimum time (find drift and/or time) |
| Difficulty | Standard +0.3 This is a standard M1 relative velocity problem using vector triangles and basic trigonometry. Part (a) involves straightforward application of tan and cos to find V and u from a right-angled triangle. Part (b) requires the cosine rule for a non-right triangle. While it's multi-step, all techniques are routine for M1 students and the question provides clear diagrams and guidance (including 'show that' scaffolding). Slightly easier than average due to its structured nature and standard methodology. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.10d Vector operations: addition and scalar multiplication3.02a Kinematics language: position, displacement, velocity, acceleration3.02e Two-dimensional constant acceleration: with vectors |
I notice that the content provided only contains repeated numbers (4) without any actual marking scheme content to clean up.
Please provide the full extracted mark scheme text that includes:
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Once you provide the complete mark scheme, I'll clean it up and convert unicode symbols to LaTeX notation as requested.4 Water flows in a constant direction at a constant speed of $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A boat travels in the water at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the water.
\begin{enumerate}[label=(\alph*)]
\item The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $74 ^ { \circ }$ to the direction of motion of the water, as shown in the diagram.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_120_164_662_488}
\captionsetup{labelformat=empty}
\caption{Velocity of the water}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_126_186_667_890}
\captionsetup{labelformat=empty}
\caption{Velocity of the boat relative to the water}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Find $V$.
\item Show that $u = 3.44$, correct to three significant figures.
\end{enumerate}\item The boat changes course so that it travels relative to the water at an angle of $45 ^ { \circ }$ to the direction of motion of the water. The resultant velocity of the boat is now of magnitude $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The velocity of the water is unchanged, as shown in the diagram below.
$$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_132_273_1493_895}
\end{center}
Velocity of the boat relative to the water\\
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_232_355_1498_1384}
Find the value of $v$.\\
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA M1 Q4}}