# Find the turn radius and turn rate of an airplane.

Find the turn radius and turn rate of an airplane, flying at a velocity of \(403.2\, km/h\) and has a lift coefficient of \(1.3\). Airplane has a wing area of \(48\,m^2\) and weighs \(100,000\,N\), and is flying at sea level.

Airplane’s turn radius is given by

\[R = \frac{{V_\infty ^2}}{{g\sqrt {{n^2} – 1} }}\]

and turn rate is \[\frac{{{V_\infty }}}{R}\]

Here, \(n\) = load factor = \((lift/weight)=L/W\)

Airplane velocity = \(403.2\,km/h = 112\,m/s\)

\[L = \frac{1}{2}{\rho _\infty }V_\infty ^2S{C_L} = \frac{1}{2} \times 1.225 \times {\left( {112} \right)^2} \times 48 \times 1.3 = 479431.68\,N\]

Therefore,\[n = \frac{{479431.68}}{{100000}} = 4.79\]

Turn radius, \[R = \frac{{V_\infty ^2}}{{g\sqrt {{{n}^2} – 1} }} = \frac{{{{\left( {112} \right)}^2}}}{{9.8\sqrt {{{\left( {4.79} \right)}^2} – 1} }} = 273.244\,m\]

Turn rate,\[\frac{{{V_\infty }}}{R} = \frac{{112}}{{273.244}} = 0.41\,rad/s\]