Discussion
Consider a general aviation airplane with weight 10 kN and a wing planform area of 15 m2. The drag coefficient of the airplane is given as \(C_{D} = C_{D_{0}} + KC_{L}^{2}\) with \(C_{D_{0}} = 0.025\) and \(K = 0.05\). For level flight at an altitude where the density is 0.60 kg/m3 and thrust 1 kN, the maximum cruise speed is
(rounded off to the nearest integer)
Answer: A
The maximum cruise speed is 87 m/s.
In level flight thrust equals drag and lift equals weight. Writing \(q\) for the dynamic pressure, \(C_{L} = W/(qS)\), so
\[ \begin{aligned} T &= qSC_{D_{0}} + \frac{KW^{2}}{qS} \\ 1000 &= 0.375q + \frac{333333}{q} \end{aligned} \]
using \(S = 15\), \(C_{D_{0}} = 0.025\), \(K = 0.05\) and \(W = 10^{4}\) N. This is a quadratic in \(q\):
\[ \begin{aligned} 0.375q^{2} - 1000q + 333333 &= 0 \\ q &= \frac{1000 \pm 707.1}{0.75} \\ q_{max} &= 2276\ \text{Pa} \end{aligned} \]
\[ \begin{aligned} V_{max} &= \sqrt{2q_{max}/\rho} \\ &= \sqrt{2 \times 2276 / 0.60} = 87\ \text{m/s} \end{aligned} \]
The smaller root, \(q = 390.5\) Pa, gives 36 m/s, the minimum speed at which 1 kN of thrust can hold level flight, which is the trap in the options. 30 m/s and 101 m/s do not satisfy the thrust-drag balance.
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