A propeller-driven aircraft of gross mass $m = 1{,}200\text{ kg}$ has an engine delivering a constant brake power $P_{\text{engine}} = 250\text{ kW}$ with propeller efficiency $\eta_p = 0.80$. At sea level, the minimum power required for steady level flight is $P_{R,\min} = 110\text{ kW}$. Assuming acceleration due to gravity $g = 9.81\text{ m/s}^2$, the maximum rate of climb $(R/C)_{\max}$ at sea level (in $\text{m/s}$) is closest to:

(A) 7.65 m/s

(B) 11.89 m/s

(C) 5.21 m/s

(D) 15.42 m/s

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles For a propeller-driven aircraft, the rate of climb $R/C$ depends directly on excess shaft power $\Delta P = P_A - P_R$: $$R/C = \frac{P_A - P_R}{W}$$ Maximum rate of climb occurs where the excess power $(P_A - P_R)$ is maximized. Since power available $P_A = \eta_p P_{\text{engine}}$ is essentially constant with airspeed for a variable-pitch propeller, maximum excess power occurs at the speed for minimum power required $P_{R,\min}$. ### 2. Analytical Formulation 1. Aircraft weight $W$: $$W = m g = 1{,}200 \times 9.81 = 11{,}772\text{ N}$$ 2. Power available $P_A$: $$P_A = \eta_p P_{\text{engine}} = 0.80 \times 250{,}000\text{ W} = 200{,}000\text{ W} = 200\text{ kW}$$ 3. Excess power $\Delta P_{\max}$: $$\Delta P_{\max} = P_A - P_{R,\min} = 200\text{ kW} - 110\text{ kW} = 90\text{ kW} = 90{,}000\text{ W}$$ ### 3. Step-by-Step Calculation $$(R/C)_{\max} = \frac{\Delta P_{\max}}{W} = \frac{90{,}000\text{ W}}{11{,}772\text{ N}} \approx 7.645\text{ m/s} \approx 7.65\text{ m/s}$$ ### 4. Distractor Forensics - **Option A (7.65 m/s):** Correct climb rate. - **Option B (11.89 m/s):** Forgot propeller efficiency $\eta_p = 0.80$ ($(250 - 110)/11.772 = 11.89\text{ m/s}$). - **Option C (5.21 m/s):** Subtracted engine power incorrectly. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Propeller aircraft climb rate is governed by *power* excess ($R/C = \Delta P / W$), whereas jet aircraft climb rate is governed by *thrust* excess ($R/C = V (T - D)/W$). - **Standard Reference:** Anderson, J.D., *Introduction to Flight*, Chapter 6.

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