| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2015 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Bearing and speed from velocity vector |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question testing basic vector concepts: finding direction from components using arctangent, writing position as r₀ + vt, and solving when the position vector satisfies a simple geometric condition (north-west means equal i and j components). All parts are routine applications of standard techniques with no problem-solving insight required, making it easier than average but not trivial due to the bearing conversion and multi-step nature. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement3.02a Kinematics language: position, displacement, velocity, acceleration |
| Answer | Marks |
|---|---|
| \(\tan \alpha = 1/3 \Rightarrow \alpha \approx 18.4°\) | M1 A1 |
| Bearing is \(288°\) (nearest degree) | A1 (3) |
| Answer | Marks |
|---|---|
| \(\mathbf{r} = (21\mathbf{i} + 5\mathbf{j}) + t(-6\mathbf{i} + 2\mathbf{j})\) | B1 (1) |
| Answer | Marks |
|---|---|
| \(21 - 6t = -(5 + 2t)\) | M1 A1 |
| \(t = 6.5\) | A1 (3) |
## Part (a)
$\tan \alpha = 1/3 \Rightarrow \alpha \approx 18.4°$ | M1 A1 |
Bearing is $288°$ (nearest degree) | A1 (3) |
## Part (b)
$\mathbf{r} = (21\mathbf{i} + 5\mathbf{j}) + t(-6\mathbf{i} + 2\mathbf{j})$ | B1 (1) |
## Part (c)
$21 - 6t = -(5 + 2t)$ | M1 A1 |
$t = 6.5$ | A1 (3) |
**Notes for Question 3(a):**
First M1 for $\arctan(\pm 2/\pm 6)$
First A1 for a correct value from their expression, usually $18.4°$ or $71.6°$
Second A1 for $288$ (nearest degree)
**Notes for Question 3(b):**
B1 for $(21\mathbf{i} + 5\mathbf{j}) + t(-6\mathbf{i} + 2\mathbf{j})$
**Notes for Question 3(c):**
M1 for equating the negative of their i-component to their j-component oe
Allow equating the components for the M mark.
First A1 for a correct equation.
Second A1 for $t = 6.5$
---[In this question $\mathbf{i}$ and $\mathbf{j}$ are unit vectors directed due east and due north respectively.]
A particle $P$ is moving with constant velocity $(-6\mathbf{i} + 2\mathbf{j})$ m s$^{-1}$. At time $t = 0$, $P$ passes through the point with position vector $(21\mathbf{i} + 5\mathbf{j})$ m, relative to a fixed origin $O$.
\begin{enumerate}[label=(\alph*)]
\item Find the direction of motion of $P$, giving your answer as a bearing to the nearest degree. [3]
\item Write down the position vector of $P$ at time $t$ seconds. [1]
\item Find the time at which $P$ is north-west of $O$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2015 Q3 [7]}}