Standard +0.8 This is a multi-part Further Maths mechanics question requiring Newton's second law with variable resistance, forming and solving a separable differential equation (v dv/dx form), and interpreting the model. While systematic, it demands fluency with F=ma in the form mv(dv/dx), separation of variables with partial fractions, and physical reasoning about limiting behavior—significantly above standard A-level but routine for FM mechanics.
A large aeroplane, of mass 360 tonnes, starts from rest at the beginning of a straight horizontal runway. The aeroplane produces a constant thrust of 980 kN and experiences a variable resistance to motion of magnitude \(\left( 80 + 0 \cdot 1 v ^ { 2 } \right) \mathrm { kN }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the aeroplane after it has travelled \(x\) metres.
(i) Find the maximum speed that the aeroplane can attain.
(ii) Show that \(v\) satisfies the differential equation
$$3600 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 9000 - v ^ { 2 } .$$
Find an expression for \(v ^ { 2 }\) in terms of \(x\).
Given that the aeroplane must achieve a speed of at least \(85 \mathrm {~ms} ^ { - 1 }\) to take off, determine the minimum length of the runway.
Explain why, according to this model, the aeroplane will not reach the speed found in (a)(i).
\begin{enumerate}
\item A large aeroplane, of mass 360 tonnes, starts from rest at the beginning of a straight horizontal runway. The aeroplane produces a constant thrust of 980 kN and experiences a variable resistance to motion of magnitude $\left( 80 + 0 \cdot 1 v ^ { 2 } \right) \mathrm { kN }$, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of the aeroplane after it has travelled $x$ metres.\\
(a) (i) Find the maximum speed that the aeroplane can attain.\\
(ii) Show that $v$ satisfies the differential equation
\end{enumerate}
$$3600 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 9000 - v ^ { 2 } .$$
(b) Find an expression for $v ^ { 2 }$ in terms of $x$.\\
(c) Given that the aeroplane must achieve a speed of at least $85 \mathrm {~ms} ^ { - 1 }$ to take off, determine the minimum length of the runway.\\
(d) Explain why, according to this model, the aeroplane will not reach the speed found in (a)(i).\\
\hfill \mbox{\textit{WJEC Further Unit 6 2019 Q1 [15]}}