Fall factor and impact force

Quite a few sources claim (inc. Petzl and Rock and Ice) claim that, contrary to what you might expect, impact force is not a function of the length of fall, but is wholly determined by the fall factor (and constants like the climber's mass and rope stretchiness.)

This can't be right.

Case one: I have tied in directly to the anchor with the rope, leaving two inches of slack. I climb two inches above the anchor, slip, and fall. Fall factor= length of fall/length of rope=4in/2in=2.

Case two: I climb 30m above the anchor without placing any gear, then fall. Fall factor=2.

There's just no way impact forces in these cases are comparable.

Am I missing something?

Its true that fall factor is not perfect. For one it doesn't account for the acceleration. But unless you are planning on bringing a bunch of measuring equipment just in case you fall, its really a pretty good system. Its not for calculating an exact force but rather an approximate one, that can be done quickly and easily without any equipment.

Also why are you planning on taking a 30m FF 2 fall?

ive taken two semesters of physics...

energy of a body falling four inches ~ 72J
energy of a body falling 60 meters ~ 42,700J

the rope has to do that much work on the falling body to stop it. the rope performs this work as it stretches. I'm gonna assume that the distances over which the two ropes stop the body are proportional to their lengths. so the 2 inch rope has to perform the workwith a tiny fraction of the stretching of the 30 meter rope. lets assume the ropes stretch 10%

rope A has to do 72J work over .2 inches. rope b, 43kJ over 3 meters. work is equal to 1/2 the product of the distance stretched and the peak force applied by the rope.

72J= half (Force) * (.2 inches in meters)
43kJ = half (force) * (3 meters)

solve em and you get about 28 kilonewtons force in each case. So yeah if the rope stretched only 10% it would definitely rip a 2 camalot anchor lol. in reality ropes stretch about 30%. physics climbers wanna check me?

Probably isn't perfect but alot of it has to do with reaching terminal velocity of a fall compared to how fast the rope has to absorb the force.

A speed of 50% of terminal velocity is reached after only about 3 seconds, while it takes 8 seconds to reach 90%, 15 seconds to reach 99% and so on.

During a fall you accelerate faster at the start vs end. So the time between a 5ft fall vs 30ft fall isn't much difference so the amount of rope out plays more of a factor than the distance of the fall.

For those of you who are interested, there is a reasonably accessible but rigorous derivation and explanation of the use of fall factor here: Taking a Whipper—The Fall-Factor Concept in Rock Climbing

The cliff notes are: Impact force is proportional to the square root of the fall factor, as well as the mass of the climber and the spring constant of the rope. Of course the acceleration of the climber due to gravity is included. However, it is an elementary derivation and second order effects such as air resistance (which results in terminal velocity), the departure of a real rope from an ideal spring, and the friction of the rope through the biners are not included.

The take away is that fall factor is a tool for explaining and understanding the basic mechanics of a roped leader fall but is not great for calculating the impact force of a real fall.

I think the thing that you are overlooking is the amount of energy that can be absorbed by your body and harness. This is small but can reduce the effective fall factor. So for example just pulling some numbers out of my nether regions suppose that the climber weighs 60 kilograms and falls 0.1 meters. The energy generated then is

E = mgh = 60*9.8*0.1 = 58.5 Joules

If your body and harness can absorb 40 Joules then the effective fall factor is reduced by a large amount. If the fall is 2 meters though the energy generated is

E = mgh = 60*9.8*2 = 1176 Joules

and the 40 Joules absorbed by your body and harness is insignificant.

ViperScale wrote:Probably isn't perfect but alot of it has to do with reaching terminal velocity of a fall compared to how fast the rope has to absorb the force. A speed of 50% of terminal velocity is reached after only about 3 seconds, while it takes 8 seconds to reach 90%, 15 seconds to reach 99% and so on. During a fall you accelerate faster at the start vs end. So the time between a 5ft fall vs 30ft fall isn't much difference so the amount of rope out plays more of a factor than the distance of the fall.

After 3 seconds of freefall you've traveled roughly 44 meters. After 4 seconds - roughly 78 meters. I think that if you have to take air resistance into account, you've got more to worry about.

Much of the reason and you'll feel more impact force on your 30 m fall vs. your 4 inch fall, has to do with spring-rate -- how stiff your 'rope' is.
Your harness, body, the knot, etc. will absorb some energy -- that is, they will soften the spring-rate. On a small fall that energy is significant. On a 30 m fall, that energy is dwarfed by the stretch of the rope. This is because the rope's spring-rate is inversely proportional to its length. A rope twice as long is half as stiff.

Edit -- I see that Emil got here first with the same point.

It's not the speed or acceleration of the fall that generates the impact force, it's the deceleration rate. As the time elapsed for an object to stop movement approaches zero, the force generated approaches infinity. In the real world as others have mentioned elements such as harness, rope slip and your body giving way all add time into the system which has a greater impact on the net force when the initial velocity is small, but less so when it is a higher velocity. So yes a 4inch fall will have a lesser force generated than a 60m fall.

Xam wrote:For those of you who are interested, there is a reasonably accessible but rigorous derivation and explanation of the use of fall factor here: Taking a Whipper—The Fall-Factor Concept in Rock Climbing

This derivation shows that (under big assumptions) the only variable on which the impact force depends, is fall factor.

The interesting part is that he defines fall factor as total fall distance including rope stretch, divided by total un-stretched rope length. As far as I know this isn't the conventional definition of fall factor. Petzl does not include rope stretch in the fall factor definition.

Much of what is written above is wholly or partially wrong or beside the point. The "Taking a Whipper" article referenced in the link is an example of poor academic scholarship, in that it fails to mention the fact that all these calculations were done 75 years ago and includes its own idiosyncratic definition of fall factor that doesn't agree with anyone else's. (Mathematically, it turns out the definition is equivalent to the standard one, but practically the definition is worthless since you would need to know the rope stretch before knowing the fall factor, but you need the fall factor in order to calculate the rope stretch.)

The notion of fall factor comes from modeling a climbing rope as a spring and applying Hooke's Law. Once that is done, you can either use the standard second-order differential equations for simple harmonic motion or just use high-school algebra and a conservation of energy argument. Many years ago I wrote up and posted both approaches (with appropriate credit given) in a document which I believe can still be found at 4sport.ua/_upl/2/1404/Stand….

Here are a few intuitions that may help with what seems to be paradoxical.

(1) The loads at any moment are determined by the tension in the rope, and this in turn (by Hooke's Law) is proportional to how much the rope has stretched relative to its resting length (eg its percentage stretch). If very different-seeming events give rise to the same maximal percentage rope stretch, then those events will produce the same maximal loads.

(2) In order to stop a fall a system must "absorb" all the fall energy, which is to say mgh, where m is the mass of the falling object, g is the acceleration due to gravity, and h is the total fall length (which in the climbing case would include rope stretch). The energy is "absorbed" by the work done in stretching the rope. When the work done in stretching the rope is equal to the total loss of potential energy (i.e. mgh) the fall is arrested and the rope tension achieves its maximum value. This description is a verbal form of the conservation of energy equation, from which the fall-factor concept emerges, as it turns out that the maximal percentage rope stretch is determined by the ratio of the height of the fall (before rope stretch) to the total amount of rope in the system.

Very roughly speaking, a short fall with little rope in the system may stretch the rope, percentage-wise, as much as a much longer fall with a lot more rope in the system, leading to the outcome of equal rope tensions for both events.

The spring model is of course only a first approximation to rope behavior, and there are many more complex models that do a better job. That said, all the more complex models have as their core element (damped) springs.

So it's not necessarily more dangerous to take a long 30 footer, versus a little 2 foot whipper?

It is possible that the anchor loads in a certain two-foot plop could be the same as for a certain 30 foot whipper, but then a host of other factors would be involved in deciding which of the two events is "more dangerous." For example, a lot of things can happen to you during a 30 foot fall that are not likely or possible in a two-foot drop.

A factor-2 fall straight onto a quickdraw can and has broken the carabiners. But this is really extreme with a fall factor around 4.

Perhaps it is also worth noting that once the rope starts sliding through a belay device under tension, the energy absorbtion mode switches from the stretching of an elastic medium to heat dissipation, and then the actual fall length matters critically.