# 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.

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 %.