| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Apparent wind problems |
| Difficulty | Challenging +1.2 This is a standard M4 apparent wind problem requiring vector equation setup and simultaneous solving. While it involves multiple steps (two scenarios, vector subtraction, solving for unknowns), the technique is routine for this module with no novel insight required. The algebraic manipulation is straightforward once the key relationship (apparent wind = true wind - cyclist velocity) is applied. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(v(3\mathbf{i} - 4\mathbf{j}) = \mathbf{v}_w - u\mathbf{j}\) | M1A1 | |
| \(\mathbf{v}_w = 3v\mathbf{i} + (u - 4v)\mathbf{j}\) | ||
| \(w\mathbf{i} = \mathbf{v}_w - \frac{u}{5}(-3\mathbf{i} + 4\mathbf{j})\) | M1A1 | |
| \(\mathbf{v}_w = \left(w - \frac{3u}{5}\right)\mathbf{i} + \frac{4u}{5}\mathbf{j}\) | ||
| \((u - 4v) = \frac{4u}{5}\) | M1 | |
| \(v = \frac{u}{20}\) | A1 | |
| \(\mathbf{v}_w = \frac{3u}{20}\mathbf{i} + \frac{4u}{5}\mathbf{j}\) | A1 | |
| Total | 7 |
## Question 1:
| Working/Answer | Marks | Notes |
|---|---|---|
| $v(3\mathbf{i} - 4\mathbf{j}) = \mathbf{v}_w - u\mathbf{j}$ | M1A1 | |
| $\mathbf{v}_w = 3v\mathbf{i} + (u - 4v)\mathbf{j}$ | | |
| $w\mathbf{i} = \mathbf{v}_w - \frac{u}{5}(-3\mathbf{i} + 4\mathbf{j})$ | M1A1 | |
| $\mathbf{v}_w = \left(w - \frac{3u}{5}\right)\mathbf{i} + \frac{4u}{5}\mathbf{j}$ | | |
| $(u - 4v) = \frac{4u}{5}$ | M1 | |
| $v = \frac{u}{20}$ | A1 | |
| $\mathbf{v}_w = \frac{3u}{20}\mathbf{i} + \frac{4u}{5}\mathbf{j}$ | A1 | |
| **Total** | **7** | |
---\begin{enumerate}
\item \hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are unit vectors due east and due north respectively]
\end{enumerate}
A man cycles at a constant speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on level ground and finds that when his velocity is $u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the velocity of the wind appears to be $v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $v$ is a positive constant.
When the man cycles with velocity $\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, the velocity of the wind appears to be $w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $w$ is a positive constant.
Find, in terms of $u$, the true velocity of the wind.\\
\hfill \mbox{\textit{Edexcel M4 2010 Q1 [7]}}