GATE 2023 AE – Question 40
An airplane weighs 10 kN, has area 15 m², drag polar $C_D=0.025+0.05C_L^2$, and flies level at density 0.60 kg/m³ with thrust 1 kN. Maximum cruise speed (nearest integer) is
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Correct answer: (A) 87 m/s
Explanation
**Level flight at maximum speed.** In steady level flight the thrust equals the drag and the lift equals the weight.
**Step 1: lift coefficient at speed $V$.**
$$C_L=\frac{W}{\tfrac12\rho V^2S}=\frac{10\,000}{0.5\times0.60\times15\times V^2}=\frac{2222.2}{V^2}.$$
**Step 2: drag.**
$$D=\tfrac12\rho V^2S\,C_D=4.5V^2\left(0.025+0.05C_L^2\right)=0.1125V^2+0.225V^2\left(\frac{2222.2}{V^2}\right)^2.$$
$$D=0.1125V^2+\frac{1.111\times10^{6}}{V^2}\ \text{N}.$$
**Step 3: set $D=T=1000$ N.**
$$0.1125V^4-1000V^2+1.111\times10^6=0\;\Rightarrow\;V^2=\frac{1000\pm\sqrt{10^6-5\times10^5}}{0.225}=\frac{1000\pm707.1}{0.225}.$$
$$V^2=7587\ \text{or}\ 1302\;\Rightarrow\;V=87.1\ \text{m/s}\ \text{or}\ 36.1\ \text{m/s}.$$
The **larger** root is the maximum cruise speed: **87 m/s** (option A). The smaller root is the minimum speed at which the thrust can balance the drag.