Editing Fall factor (section)

===Derivation of the equation===

Conservation of energy at rope's maximum elongation ''x<sub>max</sub>'' gives

:<math> mgh = \frac{1}{2}kx_{max}^2 - mgx_{max}\ ;  \      F_{max} = k x_{max}. </math>

The maximum force on the climber is ''F<sub>max</sub>-mg''. It is convenient to express things in terms of the [[elastic modulus]] ''E'' = ''k L/q'' which is a property of the material that the rope is constructed from. Here ''L'' is the rope's length and ''q'' its cross-sectional area. Solution of the quadratic gives

:<math>F_{max} = mg + \sqrt{(mg)^2 + 2mgEqf}.</math>

Other than fixed properties of the system, this form of the equation shows that the impact force depends only on the fall factor.

Using the HO model to obtain the impact force of real climbing ropes as a function of fall height ''h'' and climber's weight ''mg'', one must know the experimental value for ''E'' of a given rope. However, rope manufacturers give only the rope’s impact force ''F<sub>0</sub>'' and its static and dynamic elongations that are measured under standard [[UIAA]] fall conditions: A fall height ''h<sub>0</sub>'' of 2 × 2.3&nbsp;m with an available rope length ''L<sub>0</sub>'' = 2.6m leads to a fall factor ''f<sub>0</sub>'' = ''h<sub>0</sub>/L<sub>0</sub>'' = 1.77 and a fall velocity ''v<sub>0</sub>'' = (''2gh<sub>0</sub>'')<sup>1/2</sup> = 9.5&nbsp;m/s at the end of falling the distance ''h<sub>0</sub>''. The mass ''m<sub>0</sub>'' used in the fall is 80&nbsp;kg. Using these values to eliminate the unknown quantity ''E'' leads to an expression of the impact force as a function of arbitrary fall heights ''h'', arbitrary fall factors ''f'', and arbitrary gravity ''g'' of the form:

:<math>F_{max} = mg + \sqrt{(mg)^2 + F_0(F_0-2m_0g_0)\frac{m}{m_0}\frac{g}{g_0}\frac{f}{f_0}} </math>

Note that keeping ''g''<sub>0</sub> from the derivation of "''Eq''" based on UIAA test into the above ''F<sub>max</sub>'' formula assures that the transformation will continue to be valid for different gravity fields, as over a slope making less than 90 degrees with the horizontal. This simple undamped harmonic oscillator model of a rope, however, does not correctly describe the entire fall process of real ropes. Accurate measurements on the behaviour of a climbing rope during the entire fall can be explained if the undamped harmonic oscillator is complemented by a non-linear term up to the maximum impact force, and then, near the maximum force in the rope, internal friction in the rope is added that ensures the rapid relaxation of the rope to its rest position.<ref name=leuthaeusser>{{cite journal|url=http://www.sigmadewe.com/bergsportphysik.html?&L=1|title=The physics of a climbing rope under a heavy dynamic load|date= June 17, 2016|accessdate =2016-06-29|author=Leuthäusser, Ulrich|journal=Journal of SPORTS ENGINEERING AND TECHNOLOGY|volume=231 |issue=2 |pages=125–135 |doi=10.1177/1754337116651184|url-access=subscription}}</ref>