# 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.

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

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