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, $$\frac{{{V_\infty }}}{R} = \frac{{112}}{{273.244}} = 0.41\,rad/s$$