August 2013  Interesting Outdoor Mathematics

How do you measure the height of a bridge when alone in the backcountry? Uncle Jacko explains how nature provides what we need for mathematics and how according to one calculation, 1.4 kg of chocolate might just be an acceptable amount during a long tramp.

Friction

Evidently more than a few mathematicians and mathematics teachers go tramping, and it is more than humbling when they point out an error. In my last UJCC, I inaccurately commented that Duncan Hall had claimed a unique mix of interesting numbers arising from the Capstan Formula. Actually, I had misquoted him as instead he had noted his surprise at finding that such an unexpected and interesting relationship could be found from such an improbable source. In fact Leonhard Euler derived the first and most elegant relationship between e, i, 1 and pi back in the 18th Century, known as Euler’s formula.

Other readers doubted the veracity of my observation that the friction around a karabiner is independent of the bending radius. While there are a few caveats on the Capstan Formula, this remarkable characteristic holds. But, no, you can’t replace a standard karabiner with a strong nail in a rappel block, for instance. If you do, you may find the nail acts more like a knife than anything else. If you have nothing better to do on a hut day, a profitable line of investigation would be to measure friction with your climbing rope. This will certainly rescue you from re-reading the soup packet instructions yet again. An Internet search will reveal the formula’s limitations: there aren’t many.

Interesting Outdoor Mathematics

Having re-piqued my interest in physics with the Capstan Formula, here are some other avenues to look into during your next trip.

You can measure the height of a bridge or bluff by dropping a stone and timing it until it hits the bottom. The formula is: Distance in metres = 5 x (time taken in seconds)2. Thank you, Sir Isaac Newton. As the formula is surprisingly insensitive to air drag, the size and shape of the stone isn’t important. Test it yourself.

More esoterically, throw a pebble into a still pond and watch the waves radiate outwards. Notice that the outermost wave peak you are watching falls behind and becomes the second one. How did that happen? You don’t have to know how to derive the formulas for wave propagation to admire the completely unintuitive case of group velocity being faster than phase velocity in two dimensions. However before undertaking this experiment, make sure that the photography buffs have taken all the shots they need first, or risk becoming part of the experiment yourself.

While you are at it, notice the splash and ripples that form when the pebble hits the water. The world of Bessel functions awaits you.

Calories

When researching Safety in the Mountains, I happened on a classic British outdoors book, Mountaincraft and Leadership by Eric Langmuir (first published in 1969, with the fourth edition still in print), which gives a fancy nomograph for working out how fast and furious you can travel up and down or across the landscape, and how much energy is required. I apologise for the poor reproduction here. Personally, I think the nomograph is of somewhat academic interest: the only time I ever travelled 30 kilometres across Fiordland, climbing 3000 metres on the way, was quite a while ago and included the Southeast ridge of Mt Tutuko. However, it does give pause for reflection, which is why I have included it here.

If you really want to know how much to eat, perhaps the most pragmatic advice comes from Lynette Hartley on the Christchurch Tramping Club’s website, www.ctc.org.nz. She cites Geoff Spearpoint, who says you need 3000 to 7000 calories per day. That’s quite a wide range, but then a world of difference exists between trogging up the Waimakariri River to Carrington Hut, and pushing on to Barker Hut a few more kilometres up-valley. 

Langmuir states that 100 grams of chocolate provides approximately 500 calories or 2000 joules. Going by that, a tough trip might require a daily intake of up to 1.4 kilograms of chocolate.

Bad Trips

Epics are those trips that you just wished were over at the time but become etched into your memory afterwards. Upon reflection, it seemed to me that Epics and bad meals are both related and under-rated. For instance, while I have had many good meals in my life, the standout ones were those involving truly awful food. True, I do sort of remember a few truly great meals, but those disasters that really stand out were ones when digesting the food had all the characteristics of chewing gum (but displaced too much further along the alimentary canal), or where the cook must have outsourced from a stock and station agent! 

So it is with tramping trips when people get lost, camps get washed out, the cooker is left behind and so on. I guess that’s what is so great about tramping: not only can you have a miserable time walking in bad weather with heavy packs in the wrong direction, but when you give up for the night you can knock over half of the food that you did remember to bring to make a thoroughly inadequate and impoverished meal: this way you gain memories to last your entire remaining life.

EPERBs

Recently I learnt that an EPERB, used by an unfortunate party lost in the Ruahines, confused searchers by indicating that the trampers had moved up to seven kilometres, when in fact they had done the right thing and stayed put. My guess is that the EPERB signals bounced off rock walls above them and so satellites in different passes got wayward readings. Normally a rescue helicopter would soon clear up the confusion, but the weather was too bad for flying. The moral of the story? EPERBs in confined gorges do not necessarily work as well as those set off on open tops. A mountain radio would have solved the problem completely.

I should add that the lost party has my admiration for heading into the Hills in mid-winter. Getting lost was unfortunate, but that happens to the best of us occasionally. In spite of soaking gear, cold and lack of sleep, they were found in good condition, which proves my point that as long as you don’t suffer some serious bad luck (which sadly does happen from time to time), it is quite hard to come a gutser in the Hills. I guess that, once they got home, they got stuck into a substantial meal – even if they won’t remember what it was in two years’ time!

Ka kite,

Robin McNeill

r.mcneill@ieee.org

This column was originally published in the November 2013 FMC Bulletin. We will be regularly re-publishing a number of stories from Uncle Jacko’s Cookery Column here on Wilderlife. 

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