Editing Fall factor (section)

===Equation for the impact force and its interpretation===

When modeling the rope as an undamped [[harmonic oscillator]] (HO) the impact force ''F<sub>max</sub>'' in the rope is given by:

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

where ''mg'' is the climber's weight, ''h'' is the fall height and ''k'' is the spring constant of the portion of the rope that is in play.

We will see below that when varying the height of the fall while keeping the fall factor fixed, the quantity ''hk'' stays constant.

There are two factors of 2 involved in the interpretation of this equation. First, the maximum force on the top piece of protection is roughly 2''F<sub>max</sub>'', since the two sides of the rope around that piece both pull downward with force ''F<sub>max</sub>'' each.
Second, it may seem strange that even when ''h=0'', we have  ''F<sub>max</sub>''=2''mg'' (so that the maximum force on the top piece is approximately 4''mg''). This is because a factor-zero fall is the sudden weighting of a slack rope with weight ''mg''. The climber falls, picking up speed as the earth pulls them downward and, simultaneously, the rope stretches and pulls upward on them. The climber starts slowing down when the upward force generated by the stretching rope equals the gravitational force ''mg''. They then keep moving downward because of their momentum but now they are slowing down, not speeding up. Eventually they come to a stop and at that instant the rope is at maximum tension pulling upward on the climber by a force of ''2mg''. Because this upward force by the rope is more than the weight ''mg'' downward, the climber then yo-yo's up. The yo-yo'ing will eventually stop when the fall energy has all dissipated by frictional forces between (and within) the rope, the protection pieces and the harness.