# Fall factor

In lead climbing using a dynamic rope, the fall factor (f) is the ratio of the height (h) a climber falls before the climber's rope begins to stretch and the rope length (L) available to absorb the energy of the fall.

The climber will fall about the same height h in both cases, but they will be subjected to a greater force at position 1, due to the greater fall factor.
${\displaystyle f={\frac {h}{L}}}$

## Impact force

The impact force is defined as the maximum tension in the rope when a climber falls. Using the common rope model of an undamped harmonic oscillator (HO) the impact force Fmax in the rope is given by:

${\displaystyle F_{max}=mg+{\sqrt {(mg)^{2}+2mghk}}=mg+{\sqrt {(mg)^{2}+2mgEqf}}}$

where mg is the climber's weight, h is the fall height and k is the spring constant of the rope. Using the elastic modulus E = k L/q which is a material constant, the impact force depends only on the fall factor f, i.e. on the ratio h/L, the cross section q of the rope, and the climber’s weight. The more rope is available, the softer the rope becomes which is just compensating the higher fall energy. The maximum force on the climber is Fmax reduced by the climber’s weight mg. The above formula can be easily obtained by the law of conservation of energy at the time of maximum tension resp. maximum elongation xmax of the rope:

${\displaystyle mgh={\frac {1}{2}}kx_{max}^{2}-mgx_{max}\ ;\ F_{max}=kx_{max}}$

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 F0 and its static and dynamic elongations that are measured under standard UIAA fall conditions: A fall height h0 of 2 × 2.3 m with an available rope length L0 = 2.6m leads to a fall factor f0 = h0/L0 = 1.77 and a fall velocity v0 = (2gh0)1/2 = 9.5 m/s at the end of falling the distance h0. The mass m0 used in the fall is 80 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:

${\displaystyle F_{max}=mg+{\sqrt {(mg)^{2}+F_{0}(F_{0}-2m_{0}g_{0}){\frac {m}{m_{0}}}{\frac {g}{g_{0}}}{\frac {f}{f_{0}}}}}}$

Note that keeping g0 from the derivation of "Eq" based on UIAA test into the above Fmax 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.[1]

When the rope is clipped into several carabiners between the climber and the belayer, an additional type of friction occurs, the so-called dry friction between the rope and particularly the last clipped carabiner. Dry friction leads to an effective rope length smaller than the available length L and thus increases the impact force.[2] Dry friction is also responsible for the rope drag a climber has to overcome in order to move forward. It can be expressed by an effective mass of the rope that the climber has to pull which is always larger than the rope mass itself. It depends exponentially on the sum of the angles of the direction changes the climber has made.[2]

A fall factor of two is the maximum that is possible in a lead climbing fall, since the length of an arrested fall cannot exceed two times the length of the rope. Normally, a factor-2 fall can occur only when a lead climber who has placed no protection falls past the belayer (two times the distance of the rope length between them), or the anchor if the climber is solo climbing the route using a self-belay. As soon as the climber clips the rope into protection above the belay, the distance of the potential fall as a function of rope length is lessened, and the fall factor drops below 2.

A fall of 20 feet exerts more force on the climber and climbing equipment if it occurs with 10 feet of rope out (i.e. the climber has placed no protection and falls from 10 feet above the belayer to 10 feet below—a factor 2 fall) than if it occurs 100 feet above the belayer (a fall factor of 0.2), in which case the stretch of the rope more effectively cushions the fall.

## Fall factors above two

In falls occurring on 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.[3]