# 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.

Asked on 8th October 2019 in

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 %.