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).