Fall factor

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In climbing, the fall factor is the length of the fall divided by the length of the rope from faller to the fixed point, whether belayer or anchor. The equation looks like this

f = \frac{l}{r}

where

f = fall factor
l = length of fall
r = length of rope out

Fall factor 2 is the maximum that should be possible in a normal climbing fall, since the length of an arrested fall can't exceed two times the length of the rope. Normally, a factor 2 fall can occur only when a leader who has placed no protection falls past the belayer, or the anchor if it's a solo climb. As soon as protection is placed, the distance of the potential fall as a function of rope length is lessened, and the fall factor drops below 2.

The severity of a fall (the force generated in the system) is proportional to the square root of the fall factor, so that a factor 2 fall is considerably more serious than a factor 1 one. This can be seen by noting that the maximal force can be estimated by

E=mgl=\int_0^{\Delta} dr F \approx \Delta F_{\rm max}

where Δ is the distance over which the fall is stopped. The distance Δ can be estimated by stating that the relative expansion of the rope is proportional to the force Fmax

\frac{\Delta}{r}\propto F_{\rm max}.

Solving this equation for Δ and inserting it into the above expression one arrives at

F_{\rm max}\propto \sqrt{l/r} = \sqrt{f}

In falls occurring in a via ferrata, fall factors can be much higher. This is possible because the length of rope between harness and carabiner is short and fixed, while the distance the climber can fall depends on the gaps between anchor points of the safety cable.

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