Last weekend, struggling at the crux of her first 5.13X, Dahlia Oviatt stretched her 18 inch frame to the max to clip the “thank God” bolt 50 feet off the deck. Her first and only piece was a bomber #6 cam sitting snugly 30 feet below. The only match for the focus in her eyes and the determination in her heart was the burning in her arms. She blew the clip. In less than 2 short seconds it was over.

Dahlia yawned, held safely 5 feet from the ground. “Waaah,” she said and Tommy lowered her and took her off belay. Dahlia pulled the rope and fired the route second try, no problem.

I know what you’re thinking. Improbable. Ridiculous. Fanciful. Sesame seeds are a lovely garnish on any salad. All true. But I swear she didn’t send the first try – not even with her climbing prodigy genes.

You might also be wondering how Tommy kept Dahlia off the deck, considering she was 30 feet above her only piece, which, in turn, was only 20 feet off the ground.

That takes a little more explanation.

In the January issue of Climbing magazine, Alex Honnold, Matt Segal and Kevin Jorgeson detailed their trip to England’s Peak District, a place known for short, very dangerous routes, often with no gear above a third of the route height. The area includes the famous headpoint testpiece Gaia and employs an “E0 – E10” rating system solely devoted to diagnosing the danger of a line.

The authors mentioned use of a “ground runner” that helped the belayer pull in slack. Such a system may consist of a couple pieces at ground level, like an anchor you would build at a belay station on a multipitch climb. A long runner to avoid rope drag and a locking carabiner the rope passes through before going to the climber complete the system.

I thought this was an interesting concept and worthy of some mathematics (pronounced math-eh-mwah-tics) to understand why and when the ground runner setup is useful. I’ll try to keep the jargon to a minimum and focus on the practical results.

The basis of this discussion is centered on the following situation: A climber falls from a distance d_{2} above her last piece (p_{L}). Her first piece (p_{1}) was placed at a distance d_{1} off the ground. The first piece p_{1} may or may not be the last piece p_{L}.

When the climber peels, the belayer can do one of three things:

a. Stand at the base of the climb (traditional belay)

b. Run backward

c. Run backward with the rope through a ground runner system like I’ve described above.

These situations are depicted below.

Fig. 1. Diagrams showing relevant quantities for the three belaying techniques. F is the point of the fall. The blue line represents the rope.

In case a, the climber falls a distance 2 x d_{2}. This is the intuitive double the distance above the last piece we all think of when reconstructing the size of our whippers.

Case b is more complicated. As the belayer runs backward, she creates a triangle between her, the base of the climb and the first piece. At the moment the climber falls, the rope travels from the belayer, standing at the base of the climb, straight up to the first piece, and on to the climber. When the climber is caught, the rope travels from the belayer, now back a distance r from the base of the climb, diagonally to the first piece.

The amount of rope the belayer pulls in equals the distance from the base of the climb to the first piece, d_{1}, subtracted from the length of the hypotenuse of the triangle, Z. Assuming the route is near vertical, the triangle will have a right angle and the Pythagorean Theorem applies. The hypotenuse is thus Z = (d_{1}^{2} + r^{2})^{1/2} and the amount of rope the belayer pulled in is therefore Z – d_{1} = (d_{1}^{2} + r^{2})^{1/2} – d_{1}. Finally, the distance the climber falls is (2 x d_{2}) – (Z – d_{1}) = (2 x d_{2}) + d_{1} – (d_{1}^{2} + r^{2})^{1/2}.

In case c, as the belayer runs backward, she pulls in a distance of rope r. The climber now falls (2 x d_{2}) – r.

To this point I have ignored rope stretch, but will include it later.

So far so good. But here’s the big monkey wrench: In the above equations, the distance the belayer runs backward, r, depends on the amount of time he has to run i.e. the amount of time the climber is in the air. The distance (and time in which) the climber falls depends on the amount the belayer can run backward. In other words, you can’t simply figure out how far the climber falls and then subtract some constant amount due to the belayer. The two quantities are intertwined.

Taking case c as an example, the climber falls and the belayer immediately begins running backward. The distance the climber falls is determined by how far the belayer can run. But how far the belayer can run depends on how much time he has to run, i.e. how much time the climber is in the air.

The trick is to determine how long the climber will be in the air taking into account the running belay.

When the climber peels, he accelerates downward at a rate of 9.8 meters/s^{2} or 32 feet/s^{2}. This is a fundamental measure of gravity on Earth and is known as the gravitational constant, g. He will fall a total distance of ½ x g x t^{2}. If we know time, we know how far he falls (Fig. 2).

But how fast can someone run backward? I decided to find out in an elaborate experiment involving way too much spandex. With Charlie’s ultra-precise cell-phone stopwatch and eyeball accuracy, I took to the streets to do some backward running.

Starting from a standstill, I can run backward about 10 feet in the first second, 13 feet in the second and 16 feet in each subsequent second. There is a short period of acceleration, followed by a constant speed. To avoid finding the roots of a nasty depressed quartic equation, I will ignore the period of acceleration and estimate that someone can run backward at an instantaneous and constant rate of 10 feet/s, or r = 10 x t. Given that the terrain of each climbing area will be different, the variability in how fast someone can run ensures this simplification will not practically limit the results.

The governing equations for time are thus:

½ x g x t^{2} = (2 x d_{2}) + d_{1} – [d_{1}^{2} + (10 x t)^{2}]^{1/2} (case 2)

½ x g x t^{2} = (2 x d_{2}) – 10 x t (case 3).

I won’t bore you further with how to solve these equations. Enjoy the video below for a little taste of the mathematical guts.

OK, now on to the fun stuff. What does all this mean? Let’s look at a few situations. Keep in mind that we’re interested in making scary routes a little less dangerous – we’re not talking about 8 foot whippers.

Climbing ropes have a maximum stretch of around 6% – 9%. I’ve factored in a rope stretch of 4% for all of the following data points.

A. The climber’s first and only piece is at 10 feet. This situation is shown graphically for all three cases in Fig. 3.

One can see that as expected, the traditional belay results in a deck when the climber is at 19 feet (remember rope stretch). The running belay without a ground runner results in a deck when the climber falls from about 25 feet. The use of a ground runner extends this to a deck from 32 feet, a full 13 feet more than the traditional belay and at a distance more than three times the height of the first and only piece.

B. The climber’s first and only piece is at 20 feet. This is shown in Fig. 4.

In this situation, the traditional belay results in a deck from 38 feet. With a running belay you can buy an extra 6 feet and would deck at 44 feet. The ground runner will allow you to climb to 56 feet without decking.

Note that in situation A with the first piece at 10 feet the ground runner allows the climber to climb to 3.2 times d_{1} before decking, while in situation B with the first piece at 20 feet, the climber can only advance to 2.8 times d_{1} before decking. This is a reflection of the fact that the entire time the climber is in the air, he is accelerating. In other words his speed is constantly increasing. Meanwhile, the belayer is traveling, to first approximation, at a constant speed. In general, for belaying cases 2 and 3, the longer the climber falls, the greater the amount of slack the belayer can take in, but with diminishing effectiveness.

It is worth note that in situations A and B (Figs. 3 and 4), an addition of a piece or pieces of protection above the first piece would shift the deck line up by the distance between the first and last pieces. The slope of the line will not change.

C. The climber always falls from 20 feet above her last piece, but the first piece is placed at different heights. This is shown in Fig. 5

In this situation the traditional belay results in a constant fall of 42 feet. Because falls associated with the ground runner setup do not depend on the distance to the first piece, d_{1}, they are also constant, but much less at 28 feet. However there is great variation in the falls with a simple running belay depending on the height of the first piece. At small values of d_{1}, most of the distance the belayer covers results in lengthening the hypotenuse of the triangle and therefore reducing the distance of the fall. However, when d_{1} is large, corresponding to a high first piece, most of the belayer’s efforts are wasted. The length of the hypotenuse increases very little by increasing distance away from the rock.

In summary, the use of a ground runner always outperforms a simple running belay. Obviously, in many, if not most climbing areas, it is not possible to give any type of running belay. However, in many instances (Wall Street, the Peak District, much of Vedauwoo etc.) such belays are possible. A further advantage of the ground runner is that it will not cause unnecessary tension on the first piece or risk the climber falling on the rope. Either type of running belay is most effective when trying to keep someone off the deck on a short, runout climb with low protection.

With an active belay, there is a real danger to the belayer. Besides stubbing a toe, when catching a particularly long fall, the belayer may run a decent distance from the base of the climb. The force of catching the climber can lead to being violently pulled into the rock, or the climber himself. As always, climbing is an exercise in risk management. Use your best judgement.

If you have any questions, feel free to contact me.

This video shows what solving the equations above actually looks like. Thanks to Sir Ron Propri for vital help in photography and video editing.

The mathematics of belaying from Adam Scheer on Vimeo.

by Adam

Kate - I was there, so I can totally vouch for the ground runner system. I try not to be that overprotective mom, but when Dahlia first told me of her plans for climbing her first 5.13X, I have to admit I was a little nervous. I told her, ‘Honey, I know you’ve been training to send Realization, but the protection is a little sketchy on this one’. Luckily, Adam overheard this conversation and told me there was no reason to worry. He then did some very simple calculations, which I of course grasped immediately, it was clear that I had nothing to worry about. Look for Dahlia’s climb in the next issue of Climbing.February 18, 2010 – 11:04 am

Chels - Thanks for covering the basics of climbing….when do you get to the complicated things?February 18, 2010 – 11:18 am

Eli - Damn, Adam, this is really interesting. Thanks for your research and work on this. I wasn’t familiar with this belay technique, so it’s nice to hear about.

Glad Dahlia is killin’ it already.February 18, 2010 – 11:20 am

Ron - Well done, Adam. I don’t like to be ‘that guy’ who always has to point these things out, but I think you made a mistake on the second page. That t in the denominator of the third term in the depressed quartic equation, just after you factored out the sqrt((d1^2 + d2^2)) terms, should have been a t^2. No big deal though, I’m sure fewer than 10% of people watching will even notice it.February 18, 2010 – 11:35 am

kris - Great analysis, Adam. Very good information to ponder and put to use if needed. Thanks a lot!

One other variable that is tough to account for mathematically is the human element. An interesting discussion of an example where “Use your best judgement” can be tough for the belayer is on James Pearson’s blog (the second half of the post).

A video of the route he is discussing can be seen here (and they use the ground runner setup!).

Dahlia, wow!February 18, 2010 – 3:48 pm

Adam - Thanks for bringing those links to my attention, Kris. If I would have known about those examples I would have included them in the text for people to check out. That grit climbing area in England seems like your kind of thing. Beautiful, free standing, weathered rock formations everywhere.

As far as Dahlia is concerned, I can’t wait until she’s heavy enough to belay me. She’ll be putting up topropes for me all the time.February 18, 2010 – 5:29 pm

Tyler - What the hell is math?February 18, 2010 – 6:50 pm

Jonger - Hey, Adam. This video is awesome!!!

I am very impressed of your writing down the math equation without hesitation and your very, very, very clear handed-writing. Haha.. February 19, 2010 – 8:19 am

E.F.R. - So interesting to see how you reduced the dangling participle trajectory with your use of consonants within a numerical sequence. Your interdisciplinary rhetoric is fascinating, and your penmanship keen. That being said, I do have one question: Do you really think this display of mathematical prowess will help you to pick up chicks, or do you have to run backwards to do that, too?

P.S. Dahlia is a rock goddess!February 19, 2010 – 9:28 am

lizzil - Being an Oviatt, this is no surprise that just a few weeks out of the womb Dahlia is already sending 5.13x. And she still looks like the sweetest baby ever!February 19, 2010 – 1:50 pm

Jon - Equations make Jon’s head hurt. JON SMASH!February 22, 2010 – 1:14 pm

E.F.R. - I was so busy being a dork, I forgot to tell you how cool the video is. I hope you do more creative work like this, Adam. Awesome!February 22, 2010 – 9:29 pm

William J Oviatt - I love the detailed explanation for those physical minds. Had this not been accomplished, I wouldn’t really have been able to predict the whip distance. Baby D bout pulled me straight out of my boots when she took a 50 ft run on me leading a 5.15a in her flip flops. Had the shoes not thrown her off she’d have flashed it quite easily. Adam, thank you for the wisdom…April 28, 2010 – 10:49 pm

Paul Krums - Imagine attaching a pulley to the ground-runner and a pulley to the belayer’s harness, creating a 3-to-1 system so that the belayer takes in rope 3 times the distance he runs. This pulley system would increase the possible climber height/pro height ratio to unheard of levels.November 4, 2010 – 9:41 pm

Adam - Here’s an interesting video that shows a variation of a ground runner. They use a gri gri anchored to a rock as a belay point. If the climber peels, one (of the two) belayers can run backward and take in slack without worrying about being pulled back into the rock. Not to mention that this looks like an amazing (and scaaary) climb.

June 22, 2011 – 4:39 pm