3 An aeroplane is taking off from a runway. It starts from rest. The resultant force in the direction of motion has power, $P$ watts, modelled by

$$P = 0.0004 m \left( 10000 v + v ^ { 3 } \right) ,$$

where $m \mathrm {~kg}$ is the mass of the aeroplane and $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity at time $t$ seconds. The displacement of the aeroplane from its starting point is $x \mathrm {~m}$.

To take off successfully the aeroplane must reach a speed of $80 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ before it has travelled 900 m .\\
(i) Formulate and solve a differential equation for $v$ in terms of $x$. Hence show that the aeroplane takes off successfully.\\
(ii) Formulate a differential equation for $v$ in terms of $t$. Solve the differential equation to show that $v = 100 \tan ( 0.04 t )$. What feature of this result casts doubt on the validity of the model?\\
(iii) In fact the model is only valid for $0 \leqslant t \leqslant 11$, after which the power remains constant at the value attained at $t = 11$. Will the aeroplane take off successfully?

\hfill \mbox{\textit{OCR MEI M4 2006 Q3 [24]}}