| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2018 |
| Session | October |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | When is one object due north/east/west/south of another |
| Difficulty | Moderate -0.3 This is a straightforward M1 mechanics question involving basic vector operations: finding speed from velocity (magnitude), writing position vectors using r = r₀ + vt, understanding 'due east' means equal j-components, and solving simultaneous equations for interception. All parts use standard techniques with no conceptual challenges, making it slightly easier than average A-level material. |
| 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 displacement3.02a Kinematics language: position, displacement, velocity, acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Speed \(= \sqrt{4^2 + 5^2} = \sqrt{41}\) or \(6.4031\ldots\) m s\(^{-1}\) | M1A1 | M1 for attempt to find magnitude of velocity; A1 for 6.4 or better |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((\mathbf{r} =)(3\mathbf{i} - 2\mathbf{j}) + t(4\mathbf{i} + 5\mathbf{j})\) | M1A1 | M1 for attempt at position vector with correct structure \(\mathbf{r}_0 + t\mathbf{v}\); A1 for correct expression |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{j}\) comp \(= 6\) | ||
| \(5T - 2 = 6\) | M1 | M1 for equating \(\mathbf{j}\) component of \(\mathbf{r}\) to 6, must be of form \(a + bT = 6\) |
| \(T = \frac{8}{5}\ (= 1.6)\) | A1 | A1 for 1.6 oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(t = 1.6 \Rightarrow (\mathbf{r} =)(3 + (4 \times 1.6))\mathbf{i}\ (+6\mathbf{j})\) | M1A1ft | First M1 for substituting their \(T\) into \(\mathbf{i}\) component of (b); First A1 ft with or without \(\mathbf{i}\) |
| boy travels \(9.4 - 1 = 8.4\) m | A1 | Second A1 for 8.4 or 8.4\(\mathbf{i}\) cao |
| \(\frac{8.4}{1.6}\) or \(\frac{8.4\mathbf{i}}{1.6}\) | DM1 | Dependent on first M1; divide distance/vector by \(T\ (> 0)\) to find \(v\). Note \(9.4/T\) is DM0 |
| \(v = 5.25\) | A1 | cao |
# Question 6:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Speed $= \sqrt{4^2 + 5^2} = \sqrt{41}$ or $6.4031\ldots$ m s$^{-1}$ | M1A1 | M1 for attempt to find magnitude of velocity; A1 for 6.4 or better | (2) |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(\mathbf{r} =)(3\mathbf{i} - 2\mathbf{j}) + t(4\mathbf{i} + 5\mathbf{j})$ | M1A1 | M1 for attempt at position vector with correct structure $\mathbf{r}_0 + t\mathbf{v}$; A1 for correct expression | (2) |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{j}$ comp $= 6$ | | |
| $5T - 2 = 6$ | M1 | M1 for equating $\mathbf{j}$ component of $\mathbf{r}$ to 6, must be of form $a + bT = 6$ |
| $T = \frac{8}{5}\ (= 1.6)$ | A1 | A1 for 1.6 oe | (2) |
## Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $t = 1.6 \Rightarrow (\mathbf{r} =)(3 + (4 \times 1.6))\mathbf{i}\ (+6\mathbf{j})$ | M1A1ft | First M1 for substituting their $T$ into $\mathbf{i}$ component of (b); First A1 ft with or without $\mathbf{i}$ |
| boy travels $9.4 - 1 = 8.4$ m | A1 | Second A1 for 8.4 or 8.4$\mathbf{i}$ cao |
| $\frac{8.4}{1.6}$ or $\frac{8.4\mathbf{i}}{1.6}$ | DM1 | Dependent on first M1; divide distance/vector by $T\ (> 0)$ to find $v$. Note $9.4/T$ is DM0 |
| $v = 5.25$ | A1 | cao | (5) **[11]** |
---6. The point $A$ on a horizontal playground has position vector $( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }$. At time $t = 0$, a girl kicks a ball from $A$. The ball moves horizontally along the playground with constant velocity $( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
Modelling the ball as a particle, find
\begin{enumerate}[label=(\alph*)]
\item the speed of the ball,
\item the position vector of the ball at time $t$ seconds.
The point $B$ on the playground has position vector $( \mathbf { i } + 6 \mathbf { j } ) \mathrm { m }$. At time $t = T$ seconds, the ball is due east of $B$.
\item Find the value of $T$.
A boy is running due east with constant speed $\nu \mathrm { ms } ^ { - 1 }$. At the instant when the girl kicks the ball from $A$, the boy is at $B$.
Given that the boy intercepts the ball,
\item find the value of $v$.
\includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-23_68_47_2617_1886}
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2018 Q6 [11]}}