| Exam Board | AQA |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2005 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Relative velocity: find resultant velocity (magnitude and/or direction) |
| Difficulty | Standard +0.3 This is a standard M1 relative velocity problem requiring vector addition with given speeds and angles. Part (a) uses sine rule to find an angle (show-that format reduces difficulty), and part (b) applies sine or cosine rule to find resultant magnitude. Straightforward application of triangle methods with no novel insight required, slightly easier than average A-level. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05b Sine and cosine rules: including ambiguous case1.10a 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 |
|---|---|---|
| \(\frac{\sin60°}{6} = \frac{\sin\alpha}{2}\) | M1, A1 | Use of sine rule; Correct LHS |
| \(\alpha = 16.8°\) | A1, A1 (4) | Correct RHS; AG correct \(\alpha\) from correct working |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{v}{\sin(180°-60°-16.8°)} = \frac{6}{\sin60°}\) | M1, A1 | Use of sine rule to find \(v\); Correct equation |
| \(v = 6.74 \text{ or } 6.75 \text{ ms}^{-1}\) | A1 (3) | Correct \(v\) |
## Question 6:
### Part (a)
$\frac{\sin60°}{6} = \frac{\sin\alpha}{2}$ | M1, A1 | Use of sine rule; Correct LHS
$\alpha = 16.8°$ | A1, A1 (4) | Correct RHS; AG correct $\alpha$ from correct working
### Part (b)
$\frac{v}{\sin(180°-60°-16.8°)} = \frac{6}{\sin60°}$ | M1, A1 | Use of sine rule to find $v$; Correct equation
$v = 6.74 \text{ or } 6.75 \text{ ms}^{-1}$ | A1 (3) | Correct $v$
---6 A motor boat can travel at a speed of $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the water. It is used to cross a river in which the current flows at $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resultant velocity of the boat makes an angle of $60 ^ { \circ }$ to the river bank, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_561_1339_1692_350}
The angle between the direction in which the boat is travelling relative to the water and the resultant velocity is $\alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\alpha = 16.8 ^ { \circ }$, correct to three significant figures.
\item Find the magnitude of the resultant velocity.
\end{enumerate}
\hfill \mbox{\textit{AQA M1 2005 Q6 [7]}}