A propeller airplane initially weighs 4905 N, has wing area 6.84 m² and maximum $C_L^{3/2}/C_D=13$. Propeller efficiency is 0.9 and specific fuel consumption is 0.76 kg/kWh. At density 1.225 kg/m³ and g=9.81 m/s², find the fuel weight in N required for 18 h endurance (nearest integer).
For maximum endurance, aerodynamic power is $P_a=\sqrt{2/(\rho S)}W^{3/2}/13$ W and shaft power is Pₐ/0.9. With time in hours, fuel mass loss gives $dW/dt=-aW^{3/2}$, where $a=g(0.76)/(1000\times0.9\times13)\sqrt{2/(1.225\times6.84)}$. Integrate to obtain $W_f^{-1/2}=W_i^{-1/2}+a(18)/2$. This gives Wf=3427.72 N and fuel weight 4905−3427.72=1477.28 N, or 1477 N. This accounts for the decreasing required power as fuel burns; using the initial power for all 18 h would overestimate the fuel.
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