Hillplodder’s Laws of Motion

I’m going to let you into a little secret – I did Maths at university and then went on to become an accountant.  So it’s probably no surprise to any readers of this blog that there’s a fair amount of analysis of my hill walking contained within.

Now, I’d like to emphasise that it’s not just for the sake of it, driven by some geeky obsession with figures (ok, maybe a little bit), but because for a while now it has seemed to me that I walk to a pattern.  And if I can work out what that pattern is, I can bring more predictability to my walk planning so that I never miss the last bus or get overtaken by sunset.  So this post is an attempt to share the results of my experiments, and to give my equations of motion.

So for the last 18 months or so, I’ve been taking a GPS-enabled watch on my walks and this tracks my distance, altitude, height gained and lost, time and also gives me an estimate of calories burned (a rather flattering one, because I believe I’m fitter than my weight would suggest).  What this means, is that I have built up some data which I’ve looked at to try to find the magic formula for how I walk.

It’s fair to say, I’ve not been 100% successful, and although it’s easy to see some broad patterns in the data, it’s also clear that there are many factors that affect my walking – such as the weather, trip fitness, terrain, issues encountered on the walk and, of course, how heavily laden I am.  Let’s look at some of these.

Weather: easy – the wetter and windier the weather, the slower the walk.  But the weather seems to interact with other factors (such as terrain) so that it isn’t really possible to boil it down to a number. This isn’t much use when planning a walk, especially as I can’t always know what the weather will be in advance.  So I just make sure I allow extra time on top of the walk time I’ve calculated.  Of course, if the weather is really unpleasant, I’m just as likely to bail out early.

Trip Fitness: on a multi-day walking trip, I get a bit fitter as the trip progresses, but early on I suffer from tiredness if I do too much too early.  So I always try to craft a trip into this rough shape: a warm-up walk on the first day (usually easy as I am travelling and only have half a day anyway), a normal length walk on day 2, then an easier third day, before going back to a harder 4th day.  After that, I still try to alternate easier and harder days but the difference between them shrinks.  One way I achieve this is by planning treks where I spend two nights in the same place and have one heavily laden day walking in, a light day in between, before moving on fully laden on the third day.  But despite understanding this pattern, I haven’t found a way to accurately quantify it into a factor, and have had to settle for broad guidelines in terms of suitable limits of distance and ascent for the different types of days.

Terrain: there’s definitely a difference between what I call normal countryside walking, such as that locally, and proper hill walking, such as in the Lakes.  In normal countryside where the gradients are shallower, I can often walk almost as fast uphill as on the flat.  The terrain underfoot also tends to be more predictable and less likely to slow me down.  In the hills however, there’s much more variation and any calculation of walk time including ascent really needs to factor in the ground underfoot.  This isn’t always practical, but I’ve found that I have to up my walk time by around 20% compared with the equivalent walk in “normal” countryside.

Issues: A walk plan can’t completely foresee where on a walk you will go wrong.  A wrong turning that I realise I’ve taken half a mile down the wrong side of a hill could mean a rough contour around the hill, or a re-ascent, or even the curtailment of a walk.  I can’t really plan for these other than to have contingency time and a range of exit strategies.  A couple of examples from my Lake District trip in July.  Firstly the walk over from Ennerdale to Wasdale via Steeple, Haycock and Seatallan.  I got to Haycock, tried to contour around from Little Gowder Crag to the path to Seatallan and found it hard going at a time I was beginning to tire (and on a heavily laden day).  I abandoned Seatallan itself and instead opted to walk out by Greendale Tarn.  What I didn’t foresee was the time it would take to cross the boggy ground around the tarn.  None of my formulas worked for this situation.  Similarly, on the next day when I walked from Wasdale to Borrowdale via Lingmell and the Corridor Route, I somehow lost the path and ended up below Stand Crag and had to contour roughly to get back around to Sty Head.  I lost loads of time, and part of the loss is caused by worry when you’re in the situation.

Load: I’m close to working this one out.  Clearly, the heavier the load, the slower the walk.  But it doesn’t make much difference on gentler terrain.  So as a result I tend to over-estimate the time for normal countryside walks when using my standard multipliers for distance and ascent.  But then on hill walks, other factors combine with load to make it less predictable.  All I know is that my ascent speed falls by 25%-50% depending on the terrain, weather etc.

The Laws of Motion

Having looked at what I can’t quantify, here’s what I can.  Now most people reading this will have heard of Naismith’s Rule – that walk time can be estimated at 1 hour per 3 miles of distance, plus an hour extra for each 2,000ft of ascent.  By comparing these two hourly amounts, this effectively says that climbing the same amount as a walk on the flat takes about 8 times longer.  So to calculate walk time you take the distance, then add on 8 times the ascent (in the same units as distance) and divide the result by your flat speed (expressed in the same unit scale as the rest).  So for example a walk of 20km with 1,600m of ascent and a normal flat speed of 4.8 kph should take 6 hours 50 mins.

Of course, Naismith’s rule is only a guess and won’t suit everyone.  Indeed, I’ve seen it expressed in several different ways and often with slightly different figures. It doesn’t factor in descent or the walker’s fitness, other than via the normal flat speed.  Personally, I find I have a good flat speed, but ascent takes me longer than Naismith’s rule calculates.  Various people have tried to come up with adjustments to Naismith’s rule to allow for these, and I’m no exception.

Hillplodder’s First Law of Motion

A descent along the same route as an ascent will take 2/3 of the ascent time.

This is actually the first rule I ever came up with, and pre-dates any deliberate measurement.  I just noticed it on my Lake District trip in 2006, used it for the rest of the trip and found it worked.  It’s a simple rule – descending takes 2/3 of the time it took to ascend.

An example where this worked perfectly was an early morning walk along Scandale and climbing up towards the pass, starting from Ambleside just before 6am one morning.  I had to be back for 8:30 to catch breakfast at the B&B and with 2.5 hours available, made myself stop and turn around at 7:30, and arrived back at the B&B almost bang on 8:30.

The rule still broadly works where the return route is different to the outward route, although it becomes a lot less precise.  This is partly because of lack of familiarity with the terrain and the need to find the route.  I use this in combination with my distance and height gain rules to estimate walk times, and whilst on a walk, this rule is invaluable in helping me work out my turnaround time, or point of no return.  That is, the point at which I have to stop walking away from the end point and walk towards it to catch the last bus or beat sunset, or whatever the deadline is.

Hillplodder’s Second Law of Motion

500ft of ascent can be simulated by 1 mile of flat walking.

Actually, I borrowed this one from an article in Trail.  The article was about maintaining hill fitness, and suggested how much extra distance you needed to add onto your walk or run if you live somewhere really flat, like I do.

Since I walk pretty consistently at a flat speed of 3 miles per hour, this law can also be expressed as:

500ft of ascent takes 20 minutes.

This has proved to be pretty accurate when hillwalking with a normal day pack.  But I have to adjust this when I’m more heavily laden. Interestingly, I have recently found that when walking on flatter terrain (the situation this rule was born from), the 150m of ascent this is equivalent to should probably be more like 200m.

Hillplodder’s Third Law of Motion

A heavy pack reduces speed by approximately 10%.

This came out of my Lake District trip in July this year, and by comparing to the other trips I’ve done during the year, using a variety of pack weights and in a range of conditions.  However, in order for this law to hold the following condition must be satisfied: the walk must be done using Pacerpoles. These are essential for maintaining momentum and a consistent speed, which would drop off more on a heavy day without them.

It is important to note that this law also only takes account of pack weight differential between two identical walks.  Further adjustments would need to be made for different terrain or weather.  And it also assumes that the length of the walk is within realistic limits of what I know I can do.

In Conclusion

So there you have Hillplodder’s first 3 Laws of Motion, and apologies for the nod to Sir Isaac Newton, which I couldn’t resist given my background.  I am still working on the overall Law that brings all of the above together to create a single formula for predicting a walk time. So far it looks like this:

Hillplodder's Rulewhere:

t = estimated walk time

f = flat distance

c = height climbed

d = height descended

s = flat walking speed

But as I refine the calculation to take effect of the subtleties that affect a walk, ultimately, it will look something like this:

Hillplodders Enhanced Rulewhere t, f, c, d and s are as defined above, and:

TA = terrain adjustment (currently I add 20% for a hillwalk and nothing for a normal countryside walk)

WA = weather adjustment

FA = fitness adjustment

IA = issues adjustment

LA = load adjustment

At the moment the simple form of the formula above over-estimates slightly a normal countryside walk, and underestimates a hillwalk.  I have found by empirical experiment that adding about 20% for a hillwalk evens most of this out.

I should, of course, also caveat everything above, by making it clear that the figures above reflect my walking capabilities (or lack of!) and you, dear reader, will likely walk to a different pattern to me.  So I’d love to hear anyone else’s thoughts and what you work to.


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