| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2006 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Interception: verify/find meeting point (position vector method) |
| Difficulty | Moderate -0.8 This is a standard M1 vectors question testing routine techniques: speed from velocity magnitude, bearing from angle calculation, collision by equating position vectors, and distance using unit vectors. All parts follow textbook methods with no novel problem-solving required, making it easier than average A-level questions overall. |
| 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 multiplication1.10e Position vectors: and displacement1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Speed of \(A = \sqrt{1^2 + 6^2} \approx 6.08 \text{ m s}^{-1}\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Bearing \(\approx 351\) | M1 A1 A1 | (3 marks) |
| Answer | Marks |
|---|---|
| (E.g.) i components equal ⟹ \(2-t = -26+3t\) ⟹ \(t = 7\) | B1 (either) M1 A1 ↓ M1 |
| i components at \(t=7\): \(A: -10+6t = 32\) | A1 cso |
| Answer | Marks |
|---|---|
| Same, so collide at \(t = 7\) s at point with p.v. \((-5\mathbf{i}+32\mathbf{j})\) m | (5 marks) |
| Answer | Marks |
|---|---|
| New velocity of \(B = \frac{8}{5}(3\mathbf{i}+4\mathbf{j}) \text{ m s}^{-1}\) | B1 |
| p.v. of \(B\) at \(7\) s \(= -26\mathbf{i}+4\mathbf{j}+1.6(3\mathbf{i}+4\mathbf{j}) \times 7 = 7.6\mathbf{i}+48.8\mathbf{j}\) | M1 A1 ↓ M1 ↓ |
| \(\overrightarrow{PB} = \mathbf{b}-\mathbf{p} = 12.6\mathbf{i}+16.8\mathbf{j}\) (in numbers) | M1 A1 |
| Distance \(= \sqrt{12.6^2 + 16.8^2} = 21 \text{ m}\) | (6 marks) |
## Part (a)
Speed of $A = \sqrt{1^2 + 6^2} \approx 6.08 \text{ m s}^{-1}$ | M1 A1 | (2 marks)
## Part (b)
$\tan \theta = 1/6$ ⟹ $\theta \approx 9.46°$
Bearing $\approx 351$ | M1 A1 A1 | (3 marks)
## Part (c)
p.v. of $A$ at time $t = (2-t)\mathbf{i} + (-10+6t)\mathbf{j}$
p.v. of $B$ at time $t = (-26+3t)\mathbf{i} + (4+4t)\mathbf{j}$
(E.g.) i components equal ⟹ $2-t = -26+3t$ ⟹ $t = 7$ | B1 (either) M1 A1 ↓ M1 |
i components at $t=7$: $A: -10+6t = 32$ | A1 cso |
$B: 4+4t = 32$
Same, so collide at $t = 7$ s at point with p.v. $(-5\mathbf{i}+32\mathbf{j})$ m | (5 marks)
## Part (d)
New velocity of $B = \frac{8}{5}(3\mathbf{i}+4\mathbf{j}) \text{ m s}^{-1}$ | B1 |
p.v. of $B$ at $7$ s $= -26\mathbf{i}+4\mathbf{j}+1.6(3\mathbf{i}+4\mathbf{j}) \times 7 = 7.6\mathbf{i}+48.8\mathbf{j}$ | M1 A1 ↓ M1 ↓ |
$\overrightarrow{PB} = \mathbf{b}-\mathbf{p} = 12.6\mathbf{i}+16.8\mathbf{j}$ (in numbers) | M1 A1 |
Distance $= \sqrt{12.6^2 + 16.8^2} = 21 \text{ m}$ | (6 marks)
**Total for Question 6: 16 marks**
---[In this question the horizontal unit vectors $\mathbf{i}$ and $\mathbf{j}$ are due east and due north respectively.]
A model boat $A$ moves on a lake with constant velocity $(-\mathbf{i} + 6\mathbf{j}) \text{ m s}^{-1}$. At time $t = 0$, $A$ is at the point with position vector $(2\mathbf{i} - 10\mathbf{j})$ m. Find
\begin{enumerate}[label=(\alph*)]
\item the speed of $A$, [2]
\item the direction in which $A$ is moving, giving your answer as a bearing. [3]
\end{enumerate}
At time $t = 0$, a second boat $B$ is at the point with position vector $(-26\mathbf{i} + 4\mathbf{j})$ m.
Given that the velocity of $B$ is $(3\mathbf{i} + 4\mathbf{j}) \text{ m s}^{-1}$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{2}
\item show that $A$ and $B$ will collide at a point $P$ and find the position vector of $P$. [5]
\end{enumerate}
Given instead that $B$ has speed $8 \text{ m s}^{-1}$ and moves in the direction of the vector $(3\mathbf{i} + 4\mathbf{j})$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{3}
\item find the distance of $B$ from $P$ when $t = 7$ s. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2006 Q6 [16]}}