| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Distance between two moving objects |
| Difficulty | Standard +0.8 This M4 mechanics question requires setting up position vectors as functions of time, finding the distance function, then minimizing it using calculus (completing the square or differentiation). Part (b) adds complexity by requiring solving a quadratic inequality. While systematic, it demands multiple coordinated steps and careful algebraic manipulation beyond standard A-level fare. |
| Spec | 1.10f Distance between points: using position vectors1.10h Vectors in kinematics: uniform acceleration in vector form3.02a Kinematics language: position, displacement, velocity, acceleration3.02b Kinematic graphs: displacement-time and velocity-time |
\begin{enumerate}
\item \hspace{0pt} [In this question the horizontal unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and due north respectively.]
\end{enumerate}
A ship $A$ has constant velocity $( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }$ and a ship $B$ has constant velocity $( - \mathbf { i } + 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. At noon, the position vectors of the ships $A$ and $B$ with respect to a fixed origin $O$ are $( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { km }$ and $( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }$ respectively.
Find\\
(a) the time at which the two ships are closest together,\\
(b) the length of time for which ship $A$ is within 2 km of ship $B$.
\hfill \mbox{\textit{Edexcel M4 2017 Q1 [8]}}