How the aircraft’s range will vary?

If an aircraft takes-off with 10 % less fuel in -comparison to its standard configuration,how its range will vary.

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    Let the aircrafts fuel weight be ‘f’ and aircraft structural weight be ‘s’.The jet aircraft’s range is proportional to

    \(\sqrt{s+f}-\sqrt{s}\)

    Therefore fractional reduction in range is

    \(\frac{\sqrt{s+f}-\sqrt{s+0.9f}}{\sqrt{s+f}-\sqrt{s}}\)

    If  \(f/s\sim 1 \)

    \(\frac{\Delta R}{R}=\frac{\sqrt{2}-\sqrt{1.9f}}{\sqrt{2}-\sqrt{1}}
    \\=8.6 \%\)

    If \(f/s<<1\) using binomial expansion  we find that  \(\frac{\Delta R}{R}\) converges to 10 %.

    For propeller aircraft the range is proportional to \(ln\left ( 1+\frac{f}{w} \right )\)

    \(\frac{\Delta R}{R}=\frac{ln\left ( \frac{s+f}{s+0.9f} \right )}{ln\left ( \frac{s+f}{s} \right )}\)

    If \(f/s\sim 1\) then reduction in range will be 7.4 %

    If \(f/s\sim 0\) using series expansion for logarithm this converges to 10 %.

    Therefore we see that in all these cases reduction in range is less than 10 %.

    Answered on 8th October 2019.
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