The GATE Grind

GATE 2023 AE – Question 17

Flight Mechanics & Space Dynamics · Airplane performance: drag polar, take-off and landing, climb and descent, ceilings, range and endurance, load factor, turning flight, V-n diagram, winds · 1 mark · Multiple choice

An ideal glider has $C_D=C_{D0}+C_{Di}$ with $C_{Di}=KC_L^2$. For maximum range, the ratio $C_{D0}/C_{Di}$ is

  1. 1
  2. $1/3$
  3. 3
  4. $3/2$

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Correct answer: (A) 1

Explanation

**Gliding range.** A glider with lift-to-drag ratio $L/D$ descending through a height $h$ covers a horizontal distance $R=h\,(L/D)$ in still air. The maximum range therefore needs the maximum $L/D$.

**Maximise $L/D$.** With the parabolic polar $C_D=C_{D0}+KC_L^2$:
$$\frac{C_L}{C_D}=\frac{C_L}{C_{D0}+KC_L^2}.$$
Setting the derivative with respect to $C_L$ to zero:
$$C_{D0}+KC_L^2-2KC_L^2=0\;\Rightarrow\;C_{D0}=KC_L^2 .$$

Since $C_{Di}=KC_L^2$, the zero-lift drag equals the induced drag at best $L/D$:
$$\frac{C_{D0}}{C_{Di}}=\mathbf{1}.$$