| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2006 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Position vector at time t (constant velocity) |
| Difficulty | Moderate -0.3 This is a standard M1 kinematics question using vectors with constant velocity. Parts (a)-(c) involve routine calculations (magnitude, bearing, position vector), while parts (d)-(f) require setting up position vectors and solving simple equations. The multi-part structure and real-world context add length but not conceptual difficulty—all techniques are textbook standard for M1. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10h Vectors in kinematics: uniform acceleration in vector form |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Speed \(= \sqrt{(2.5^2 + 6^2)} = 6.5 \text{ km h}^{-1}\) | M1 A1 | (2) |
| (b) Bearing \(= 360 - \arctan(2.5/6) \approx 337\) | M1 A1 | (2) |
| (c) \(\mathbf{R} = (16 - 3 \times 2.5)\mathbf{i} + (5 + 3 \times 6)\mathbf{j} = 8.5\mathbf{i} + 23\mathbf{j}\) | M1 A1 | (2) |
| (d) At 1400 \(\mathbf{s} = 11\mathbf{i} + 17\mathbf{j}\) | M1 A1 | |
| At time \(t\), \(\mathbf{s} = 11\mathbf{i} + (17 + 50)\mathbf{j}\) | ↓ M1 A1 | (4) |
| (e) East of \(\mathbf{R} \Rightarrow 17 + 5t = 23 \Rightarrow t = 6/5 \Rightarrow 1512 \text{ hours}\) | M1 A1 | (2) |
| (f) At 1600 \(\mathbf{s} = 11\mathbf{i} + 27\mathbf{j}\) | M1 | |
| \(\mathbf{s} - \mathbf{r} = 2.5\mathbf{i} + 4\mathbf{j}\) | ↓ M1 A1 | |
| Distance \(= \sqrt{(2.5^2 + 4^2)} \approx 4.72 \text{ km}\) | (3) |
| (a) Speed $= \sqrt{(2.5^2 + 6^2)} = 6.5 \text{ km h}^{-1}$ | M1 A1 | (2) |
| (b) Bearing $= 360 - \arctan(2.5/6) \approx 337$ | M1 A1 | (2) |
| (c) $\mathbf{R} = (16 - 3 \times 2.5)\mathbf{i} + (5 + 3 \times 6)\mathbf{j} = 8.5\mathbf{i} + 23\mathbf{j}$ | M1 A1 | (2) |
| (d) At 1400 $\mathbf{s} = 11\mathbf{i} + 17\mathbf{j}$ | M1 A1 | |
| At time $t$, $\mathbf{s} = 11\mathbf{i} + (17 + 50)\mathbf{j}$ | ↓ M1 A1 | (4) |
| (e) East of $\mathbf{R} \Rightarrow 17 + 5t = 23 \Rightarrow t = 6/5 \Rightarrow 1512 \text{ hours}$ | M1 A1 | (2) |
| (f) At 1600 $\mathbf{s} = 11\mathbf{i} + 27\mathbf{j}$ | M1 | |
| $\mathbf{s} - \mathbf{r} = 2.5\mathbf{i} + 4\mathbf{j}$ | ↓ M1 A1 | |
| Distance $= \sqrt{(2.5^2 + 4^2)} \approx 4.72 \text{ km}$ | | (3) |
**Guidance:**
- (a) M1 needs square, add, and $\sqrt{}$ correct components.
- (b) M1 for finding acute angle = $\arctan(2.5/6)$ or $\arctan(6/2.5)$ (i.e. $67°/23°$). Accept answer as AWRT 337.
- (c) M1 needs non-zero initial p.v. used + 'their 3' x velocity vector.
- (d) Allow 1st M1 even if non-zero initial p.v. not used here.
- (e) A1 is for answer as a time of the day.
- (f) 1st M1 for using $t = 2$ or $4$ (but *not* 200, 400, 6, 16 etc) and forming $\mathbf{s} - \mathbf{r}$ or $\mathbf{r} - \mathbf{s}$.\begin{enumerate}
\item \hspace{0pt} [In this question the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and north respectively.]
\end{enumerate}
A ship $S$ is moving with constant velocity $( - 2.5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. At time 1200, the position vector of $S$ relative to a fixed origin $O$ is $( 16 \mathbf { i } + 5 \mathbf { j } )$ km. Find\\
(a) the speed of $S$,\\
(b) the bearing on which $S$ is moving.
The ship is heading directly towards a submerged rock $R$. A radar tracking station calculates that, if $S$ continues on the same course with the same speed, it will hit $R$ at the time 1500.\\
(c) Find the position vector of $R$.
The tracking station warns the ship's captain of the situation. The captain maintains $S$ on its course with the same speed until the time is 1400 . He then changes course so that $S$ moves due north at a constant speed of $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Assuming that $S$ continues to move with this new constant velocity, find\\
(d) an expression for the position vector of the ship $t$ hours after 1400,\\
(e) the time when $S$ will be due east of $R$,\\
(f) the distance of $S$ from $R$ at the time 1600.
\hfill \mbox{\textit{Edexcel M1 2006 Q7 [15]}}