| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2001 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Interception: verify/find meeting point (position vector method) |
| Difficulty | Standard +0.3 This is a standard M1 vectors question involving position vectors, bearings, and basic geometry. Part (a) requires setting up equations from geometric constraints (pipeline and bearing to mast), part (b) is straightforward distance/speed calculation, parts (c-d) repeat the process with corrected information. All steps are routine applications of vector methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10e Position vectors: and displacement1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(\underline{\omega}_1 = 2\underline{j} + 6\underline{i} + 6\underline{j}\) | B1 | |
| \(= 6\underline{i} + 8\underline{j}\) | B1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \( | O\underline{\omega}_1 | = \sqrt{6^2 + 8^2} = 10 \text{ km}\) |
| \(\text{Time} = \frac{10}{5} = 2 \text{ hrs}\) | M1 A1 | (3 marks) [ft on \(\underline{\omega}_1\)] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\underline{\omega}_2 = 2\underline{j} + 6\underline{i} - 6\underline{j}\) | B1 | |
| \(= 6\underline{i} - 4\underline{j}\) | M1 A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Position vector of rescue party after 1 hour \(= 3\underline{i} + 4\underline{j}\) | M1 A1 | |
| \(R\underline{\omega}_2 = 3\underline{i} - 8\underline{j}\) | M1 A1 | |
| \(\tan\theta = \frac{3}{8} = 20.6°\) | M1 | |
| \(\Rightarrow\) Required bearing \(= 180° - 20.6°\) | M1 | |
| \(\approx 159.4°\) | A1 | (7 marks) Total: 15 |
## Question 7:
### Part (a):
$\underline{\omega}_1 = 2\underline{j} + 6\underline{i} + 6\underline{j}$ | B1 |
$= 6\underline{i} + 8\underline{j}$ | B1 | (2 marks)
### Part (b):
$|O\underline{\omega}_1| = \sqrt{6^2 + 8^2} = 10 \text{ km}$ | M1 A1 |
$\text{Time} = \frac{10}{5} = 2 \text{ hrs}$ | M1 A1 | (3 marks) [ft on $\underline{\omega}_1$]
### Part (c):
$\underline{\omega}_2 = 2\underline{j} + 6\underline{i} - 6\underline{j}$ | B1 |
$= 6\underline{i} - 4\underline{j}$ | M1 A1 | (3 marks)
### Part (d):
Position vector of rescue party after 1 hour $= 3\underline{i} + 4\underline{j}$ | M1 A1 |
$R\underline{\omega}_2 = 3\underline{i} - 8\underline{j}$ | M1 A1 |
$\tan\theta = \frac{3}{8} = 20.6°$ | M1 |
$\Rightarrow$ Required bearing $= 180° - 20.6°$ | M1 |
$\approx 159.4°$ | A1 | (7 marks) **Total: 15**7. [In this question, the horizontal unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed due east and north respectively]
A mountain rescue post $O$ receives a distress call via a mobile phone from a walker who has broken a leg and cannot move. The walker says he is by a pipeline and he can also see a radio mast which he believes to be south-west of him. The pipeline is known to run north-south for a long distance through the point with position vector $6 \mathbf { i } \mathrm {~km}$, relative to $O$. The radio mast is known to be at the point with position vector $2 \mathbf { j } \mathrm {~km}$, relative to $O$.
\begin{enumerate}[label=(\alph*)]
\item Using the information supplied by the walker, write down his position vector in the form $( a \mathbf { i } + b \mathbf { j } )$.
The rescue party moves at a horizontal speed of $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. The leader of the party wants to give the walker and idea of how long it will take to for the rescue party to arrive.
\item Calculate how long it will take for the rescue party to reach the walker's estimated position.
The rescue party sets out and walks straight towards the walker's estimated position at a constant horizontal speed of $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. After the party has travelled for one hour, the walker rings again. He is very apologetic and says that he now realises that the radio mask is in fact north-west of his position
\item Find the position vector of the walker.
\item Find in degrees to one decimal place, the bearing on which the rescue party should now travel in order to reach the walker directly.
\section*{END}
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2001 Q7 [15]}}