Edexcel AEA 2022 June — Question 5 11 marks

Exam BoardEdexcel
ModuleAEA (Advanced Extension Award)
Year2022
SessionJune
Marks11
PaperDownload PDF ↗
TopicVectors 3D & Lines
TypeKinematics with position vectors
DifficultyChallenging +1.8 This AEA question requires setting up 3D position vectors with careful geometric decomposition (using the 3-4-5 triangle for the incline angle), forming a distance-squared expression, then optimizing to find when minimum distance equals 2km. While it involves multiple steps and careful algebra, the approach is relatively standard for AEA kinematics problems—no particularly novel insight is required beyond systematic application of vectors and calculus.
Spec1.10h Vectors in kinematics: uniform acceleration in vector form3.02e Two-dimensional constant acceleration: with vectors
  1. An aeroplane leaves a runway and moves with a constant speed of \(V \mathrm {~km} / \mathrm { h }\) due north along a straight path inclined at an angle \(\arctan \left( \frac { 3 } { 4 } \right)\) to the horizontal.
A light aircraft is moving due north in a straight horizontal line in the same vertical plane as the aeroplane, at a height of 3 km above the runway. The light aircraft is travelling with a constant speed of \(2 V \mathrm {~km} / \mathrm { h }\).
At the moment the aeroplane leaves the runway, the light aircraft is at a horizontal distance \(d \mathrm {~km}\) behind the aeroplane. Both aircraft continue to move with the same trajectories due north.
  1. Show that the distance, \(D \mathrm {~km}\), between the two aircraft \(t\) hours after the aeroplane leaves the runway satisfies $$D ^ { 2 } = \left( \frac { 6 } { 5 } V t - d \right) ^ { 2 } + \left( \frac { 3 } { 5 } V t - 3 \right) ^ { 2 }$$ Given that the distance between the two aircraft is never less than 2 km ,
  2. find the range of possible values for \(d\).
\begin{enumerate}
  \item An aeroplane leaves a runway and moves with a constant speed of $V \mathrm {~km} / \mathrm { h }$ due north along a straight path inclined at an angle $\arctan \left( \frac { 3 } { 4 } \right)$ to the horizontal.
\end{enumerate}

A light aircraft is moving due north in a straight horizontal line in the same vertical plane as the aeroplane, at a height of 3 km above the runway.

The light aircraft is travelling with a constant speed of $2 V \mathrm {~km} / \mathrm { h }$.\\
At the moment the aeroplane leaves the runway, the light aircraft is at a horizontal distance $d \mathrm {~km}$ behind the aeroplane.

Both aircraft continue to move with the same trajectories due north.\\
(a) Show that the distance, $D \mathrm {~km}$, between the two aircraft $t$ hours after the aeroplane leaves the runway satisfies

$$D ^ { 2 } = \left( \frac { 6 } { 5 } V t - d \right) ^ { 2 } + \left( \frac { 3 } { 5 } V t - 3 \right) ^ { 2 }$$

Given that the distance between the two aircraft is never less than 2 km ,\\
(b) find the range of possible values for $d$.

\hfill \mbox{\textit{Edexcel AEA 2022 Q5 [11]}}
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