# Why is the minimum runway length non-linear in relation to the landing weight?

The graph below shows the landing runway length requirements for a Boeing 747-8.

(taken from the Boeing 747-8 Airplane Characteristics for Airport Planning)

Considering the fact that the kinetic energy and impulse are directly proportional to the mass, I would have expected this graph to be linear –­ but clearly that's not the case. What is causing the non-linearity here?

• Probably a function of the efficiency of the spoilers and brakes in dissipating the kinetic energy (${1\over 2}mv^2$ is non-linear with velocity itself). Could also be affected by the safety margins built into that landing distance, as those numbers do not represent the absolute minimum distance needed to stop the airplane. – casey Jul 15 '14 at 20:57
• @casey Decreasing brake performance was my first idea too, but then I would have expected a more gradual change in slope. – Ventero Jul 15 '14 at 21:09
• Kinetic energy is proportional to mass, but so is the braking force! (it's really proportional to weight, but for all practical purposes those two are proportional to each other). – Jan Hudec Jul 16 '14 at 7:36

Comparing two landings, first at mass = $m_1$, the second at mass = $m_2 =1.1 \cdot m_1$, the landing speed will be $\sqrt{\frac{m_2}{m_1}}$ higher during the second landing.