The formula
factor = height of the fall ÷ rope out · F = mg + √(m²g² + 2·mg·k·factor)
Where it comes from
What makes a fall harsh is not the metres but the factor: the height fallen divided by the rope available to stop it. Falling five metres with twenty metres of rope out is a factor of 0.25 and pulls little, because there is plenty of rope to stretch; falling two metres on one metre of rope is a factor of 2, the maximum possible in sport climbing, and pulls far harder even though the fall is less than half as long. A dynamic rope behaves like a spring, and its modulus — which manufacturers publish in kilonewtons — says how much force it gives back as it stretches. The load on the bolt is higher than on the climber, because the rope runs over it like a pulley: the pull from above and the pull from below add, minus what carabiner friction absorbs.
How to work it out by hand
- Divide the height of the fall by the metres of rope out from the belayer
- That ratio is the fall factor, between 0 and 2
- Apply the elastic model formula with the rope modulus
- For the load on the bolt, add the belayer’s share less the friction
What is worth knowing
This is a model, not a measurement. It assumes only the rope stretches and nothing else absorbs energy, whereas in a real fall the knot, the harness, the climber’s body and above all the belayer lifting off the ground absorb a great deal. Real forces are therefore well below those of the UIAA test, which uses a rigid mass on purpose to be demanding. It is for understanding why the factor matters more than the height, not for certifying that an anchor will hold: that is what standards and certified gear are for.