Question

The maximum permissible load factor and the maximum lift coefficient for an airplane are $n_{\max}=7$ and $C_{L,\max}=2$, respectively. For a wing loading $W/S=6500\ \text{N/m}^2$ and air density $\rho=1.23\ \text{kg/m}^3$, the speed yielding the highest possible turn rate in the vertical plane is .............. m/s. (round off to the nearest integer)

Hint:

"Corner speed'' gives the highest instantaneous turn rate and occurs when $n=n_{\max}$ and $C_L=C_{L,\max}$. Use $V_c=\sqrt{\dfrac{2(W/S)n_{\max}}{\rho C_{L,\max}}}$.

Answer 1

Step 1: Load factor with lift limit.
For instantaneous maneuver (pull-up), with $C_L$ limited by $C_{L,\max}$, \[ n=\frac{L}{W}=\frac{C_L q S}{W}=\frac{C_L\,\tfrac12\rho V^2}{W/S}. \] The speed for the corner point (max turn rate) occurs when both limits are met: $n=n_{\max}$ at $C_L=C_{L,\max}$.

Step 2: Solve for the corner speed.
\[ n_{\max}= \frac{C_{L,\max}\,\tfrac12\rho V_c^2}{W/S} \ \Rightarrow\ V_c=\sqrt{\frac{2(W/S)\,n_{\max}}{\rho\,C_{L,\max}} }. \] \[ V_c=\sqrt{\frac{2(6500)(7)}{1.23\times 2}} =\sqrt{\frac{91000}{2.46}} =\sqrt{36910.57}=192.33\ \text{m/s}. \] \[ \boxed{V_c\approx 192\ \text{m/s}} \]