| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Closest approach of two objects |
| Difficulty | Standard +0.3 This is a standard M1 closest approach problem with straightforward vector arithmetic. Students find displacement, minimize distance using calculus (or completing the square), requiring 2-3 routine steps with no geometric insight needed. Slightly easier than average due to its mechanical, textbook nature. |
| Spec | 1.10e Position vectors: and displacement1.10f Distance between points: using position vectors3.02b Kinematic graphs: displacement-time and velocity-time3.02c Interpret kinematic graphs: gradient and area |
| Answer | Marks | Guidance |
|---|---|---|
| (a) disp. of F rel to G \(= [(2t - 3) - 2]\mathbf{i} + [(t - 5) - 5]\mathbf{j} = (2t - 5)\mathbf{i} + (t - 5)\mathbf{j}\) | M1 A1 | |
| (b) \(d^2 = (2t - 5)^2 + (t - 5)^2\) | M1 | |
| \(= 4t^2 - 20t + 25 + t^2 - 10t + 25 = 5t^2 - 30t + 50\) | M1 A1 | |
| \(= 5(t^2 - 6t + 10) = 5[(t - 3)^2 + 1]\) | M2 | |
| min. \(d^2\) (and hence \(d\)) when \(t = 3\) | A1 | |
| (c) when \(t = 3, d^2 = 5\) | M1 A1 | |
| dist. \(= \sqrt{5} = 2.24 \text{ m}\) | A1 | (3sf) |
**(a)** disp. of F rel to G $= [(2t - 3) - 2]\mathbf{i} + [(t - 5) - 5]\mathbf{j} = (2t - 5)\mathbf{i} + (t - 5)\mathbf{j}$ | M1 A1 |
**(b)** $d^2 = (2t - 5)^2 + (t - 5)^2$ | M1 |
$= 4t^2 - 20t + 25 + t^2 - 10t + 25 = 5t^2 - 30t + 50$ | M1 A1 |
$= 5(t^2 - 6t + 10) = 5[(t - 3)^2 + 1]$ | M2 |
min. $d^2$ (and hence $d$) when $t = 3$ | A1 |
**(c)** when $t = 3, d^2 = 5$ | M1 A1 |
dist. $= \sqrt{5} = 2.24 \text{ m}$ | A1 | (3sf) | (11)5. Two dogs, Fido and Growler, are playing in a field. Fido is moving in a straight line so that at time $t$ his position vector relative to a fixed origin, $O$, is given by $[ ( 2 t - 3 ) \mathbf { i } + t \mathbf { j } ]$ metres. Growler is stationary at the point with position vector $( 2 \mathbf { i } + 5 \mathbf { j } )$ metres, where $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors.
\begin{enumerate}[label=(\alph*)]
\item Find the displacement vector of Fido from Growler in terms of $t$.
\item Find the value of $t$ for which the two dogs are closest.
\item Find the minimum distance between the two dogs.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q5 [11]}}