| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Bearing and speed from velocity vector |
| Difficulty | Moderate -0.5 This is a straightforward M4 relative velocity question requiring standard vector subtraction, magnitude calculation using Pythagoras, and bearing conversion from arctan. All steps are routine applications of well-practiced techniques with no conceptual challenges or novel problem-solving required. |
| 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 vectors1.10d Vector operations: addition and scalar multiplication |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\mathbf{v}_{AB} = (3\mathbf{i} + 4\mathbf{j}) - (6\mathbf{i} - 5\mathbf{j})\) | M1 | Subtracting velocity of B from velocity of A |
| \(= -3\mathbf{i} + 9\mathbf{j}\) | A1 | Correct vector |
| \( | \mathbf{v}_{AB} | = \sqrt{(-3)^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}\) km h\(^{-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\tan\theta = \frac{3}{9}\) or \(\frac{9}{3}\) used appropriately | M1 | Attempt at correct angle from north |
| Bearing = \(360° - \arctan\left(\frac{3}{9}\right) = 342°\) (or \(341.6°\)) | A1 | Accept \(342°\) or \(341.6°\) |
# Question 1:
**(a) Velocity of A relative to B:**
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\mathbf{v}_{AB} = (3\mathbf{i} + 4\mathbf{j}) - (6\mathbf{i} - 5\mathbf{j})$ | M1 | Subtracting velocity of B from velocity of A |
| $= -3\mathbf{i} + 9\mathbf{j}$ | A1 | Correct vector |
| $|\mathbf{v}_{AB}| = \sqrt{(-3)^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}$ km h$^{-1}$ | A1 | Accept 9.49 or better |
**(b) Direction as a bearing:**
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\tan\theta = \frac{3}{9}$ or $\frac{9}{3}$ used appropriately | M1 | Attempt at correct angle from north |
| Bearing = $360° - \arctan\left(\frac{3}{9}\right) = 342°$ (or $341.6°$) | A1 | Accept $342°$ or $341.6°$ |
---\begin{enumerate}
\item \hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively.]
\end{enumerate}
Boat $A$ is moving with velocity ( $3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and boat $B$ is moving with velocity $( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. Find\\
(a) the magnitude of the velocity of $A$ relative to $B$,\\
(b) the direction of the velocity of $A$ relative to $B$, giving your answer as a bearing.\\
\hfill \mbox{\textit{Edexcel M4 2013 Q1 [5]}}