$.(10$ points$)$ A twin engine airplane has the following characteristics $W=1000kg,$ wing area $=12m^2,$ wing span $=6m,C_{L_{\text{m}\text{a}\text{x}}}=1\mathrm{.}4.$ Each engine is capable of providing a thrust of $800$ N $.$ The engine thrust lines are $1\mathrm{.}5$ m to either side of the axis of symmetry. The aerodynamic data is given by

$C_{l_{}}=-\frac{0\mathrm{.}0271}{r}ad$

$C_{n_{}}=\frac{0\mathrm{.}0551}{r}ad$

$C_{l_A}=\frac{0\mathrm{.}0551}{r}ad$

$C_{n_A}=-\frac{0\mathrm{.}011}{r}ad$

$C_{l_R}=\frac{0\mathrm{.}0081}{r}ad$

$C_{n_R}=-\frac{0\mathrm{.}091}{r}ad$

The maximum aileron deflection is $20$ and the maximum rudder deflection is $30.$

$($a$)$ For the right engine out case, and a sideslip angle of ${}^{**}=0,$ use the uncoupled roll and yaw equations to determine a relationship between velocity and rudder deflection, velocity and aileron deflection.

$($b$)$ What is the velocity corresponding to a rudder deflection of $12?$

$($c$)$ What is the velocity corresponding to an aileron deflection of $12?$