Question

An ideal glider has drag characteristics given by \[ C_D = C_{D0} + C_{Di}, \] where \(C_{Di} = K C_L^2\) is the induced drag coefficient, \(C_L\) is the lift coefficient, and \(K\) is a constant. For maximum range of the glider, the ratio \(\dfrac{C_{D0}}{C_{Di}}\) is:

• A. 1
• B. \(\tfrac{1}{3}\)
• C. 3
• D. \(\tfrac{3}{2}\)

Hint:

For maximum glider range, the condition is \(C_{D0} = C_{Di}\). This means the parasitic drag equals induced drag at optimum lift-to-drag ratio.

Answer 1

The Correct Option is A

Step 1: Range condition for a glider.
The range of a glider depends on maximizing the lift-to-drag ratio \(\dfrac{L}{D} = \dfrac{C_L}{C_D}\). Thus, we want to maximize \(\dfrac{C_L}{C_{D0} + K C_L^2}\).

Step 2: Optimize ratio.
Define function: \[ f(C_L) = \frac{C_L}{C_{D0} + K C_L^2}. \] Differentiate with respect to \(C_L\): \[ \frac{df}{dC_L} = \frac{(C_{D0} + K C_L^2)(1) - C_L(2KC_L)}{(C_{D0} + KC_L^2)^2}. \] Simplify numerator: \[ = \frac{C_{D0} + K C_L^2 - 2K C_L^2}{(C_{D0} + K C_L^2)^2} = \frac{C_{D0} - K C_L^2}{(C_{D0} + K C_L^2)^2}. \]

Step 3: Set derivative = 0.
\[ C_{D0} - K C_L^2 = 0 \;\;\Rightarrow\;\; K C_L^2 = C_{D0}. \] But note: \(K C_L^2 = C_{Di}\). So, \[ C_{D0} = C_{Di}. \]

Step 4: Ratio.
\[ \frac{C_{D0}}{C_{Di}} = 1. \]

Final Answer:
\[ \boxed{1} \]