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Old 08-23-10, 06:57 PM   #1
kmcmoobud
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Touring Math?

This might be a stupid question, but I am going to ask it anyway.

In hiking, there is a concept known as Energy Required Miles...you add one mile to your hike for every 500' in up and down in distance. The concept being that one would finish a 10 mile flat quicker and easier that a 10 mile hike with lots of climbing.

Is there a similar concept for biking? I am planning a 8-10 day trip that will have a lot of climbing for the first 100-150 miles. I'm a little nervous about accurately planning my miles per day and wonder if there are any tips for calculating what is realistic.

Dorky, I know.

Thanks.
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Old 08-23-10, 07:02 PM   #2
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If you map out the grade, you can compute the speed you usually climb. For instance, up at 6 mph for 12 miles down at 37 for 6 miles. And always underestimate!
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Old 08-23-10, 07:28 PM   #3
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What complicates things for biking is:

1) A major component of your effort, if not THE major component of your effort, is overcoming wind resistance. Wind resistance isn't as big of a deal with the slower speeds of touring, but a solid 20mph headwind will cash your check pretty quick.

2) The composition of the hills matter a bit. If the hills are close together, you can use the momentum from one hill to coast partway up the next hill, which makes things a lot easier than if you blow all your momentum at the base of one hill and start up the next hill cold.

I'd plan the ride so that I was riding with the prevailing winds at the end of the ride. Headwinds don't make such a difference in the hills (unless you're above treeline). You should do fine if you keep within your known limits, and you can recover in the second half of the ride if you get beat up in the first half.
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Old 08-23-10, 07:28 PM   #4
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As long as you have some flexibility as to how long your cycling day is, it's really not necessary to do the math. You get there when you get there. You probably have some kind of an idea as to what your average speed is on the flats over distance. Just subtract a couple of MPH if you know it's going to be a hilly day, and/or add an hour or two to your estimated time. You got a bus to catch?
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Old 08-23-10, 08:38 PM   #5
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As long as you have some flexibility as to how long your cycling day is, it's really not necessary to do the math. You get there when you get there. You probably have some kind of an idea as to what your average speed is on the flats over distance. Just subtract a couple of MPH if you know it's going to be a hilly day, and/or add an hour or two to your estimated time. You got a bus to catch?
I totally agree.

However, if you want some statistics:

For 2010 so far (no loaded tours) my average unloaded road biking speed is 15.0 mph, with an average of 44 miles/ride and 2657 feet/ride. This doesn't really describe any particular ride I ever do, but it is my average, whatever that's worth.

For the 3 loaded tours I did since I started recording this stuff in SportTracks, I have an average speed of 10.6 mph, with 51 miles/ride and 2490 feet/ride. So, that means my loaded tours are a bit more hilly than unloaded, which isn't surprising since the unloaded number includes all the rides I do in winter when I can't climb. It also means I lose 30% of my speed by loading the bike. (eek!)

My actual speeds on tour ranged from 7.9 (5700 up, 950 down) to 13.5 (1600 up, 2500 down). 10.6 was both the average and the median speed for that trip. Other trips have been similar - 9.5 to 12 mph with the occasional outlier.

I don't really do flat, so it's hard to come up with a good number for that. I think the biggest takeaway is that your loaded speed will be quite a bit less than unloaded, and especially so for climbing.

This might be exaggerated for me, as I'm really small, so my power to weight is affected greatly by adding a touring load.

I have no idea if any of this helps, but I wrote it already, so I'm clicking the post button.
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Old 08-23-10, 10:50 PM   #6
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Yep, there is the Extrapolate Altitude Test (EAT):

It is eat one Snickers bar (medium size) per 1,000 foot gained and one pint of beer (light or dark) for each 1,000 foot loss. The concept is that you need the Snickers to replace the extra energy used going uphill and the beer is to reward you for making it to the top.

Sorry, but your question just reminded me of someone who made this his actual policy many miles ago. Those 8,000 foot drops made for happy days!

Actually, the theory is sort of a good idea. I am by no means qualified to quantify it with actual mathematics, but I would think some ratio must exist. However, you would have to factor in the wind strength/direction and the hill type as discussed above.

I have heard an unofficial rule of thumb for riding rail trails made of chat (screened gravel) is to anticipate doing about 80% of your "road mileage" for the same energy.

Cool question!
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Old 08-24-10, 12:35 AM   #7
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Hmm, the maths I'm trying to work out is the ratio between sealed road, dirt road, and dirt single track. I'm thinking something like 40% more effort for dirt road, and maybe 100% for single track. So 25km single track, 36km dirt road, or 50km sealed road are all the same effort. Just wild guesses at this stage.

Someday I plan to seriously analyse some GPS traces to get the answer to the OP's question though.
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Old 08-24-10, 03:31 AM   #8
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I'm with the majority here, the other factors (especially wind) make it unrealistic to attempt a precise calculation. However, my own figures bear out what valygrl has posted. Fully loaded (c.50lbs of gear) I'd expect to average about 13mph on the flat in the absence of serious headwinds. On a recent days touring, with that sort of load, I averaged 11.6 mph (moving speed, excluding stops) for an 83 mile day which included 4150 feet of climbing. That implies that every 100 feet of ascent added about 1 minute to my ride time.

Interesting way of looking at it, I hadn't thought of doing so before. Your mileage will most definitely vary, though.
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Old 08-24-10, 04:02 AM   #9
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The difference between hiking and bicycle touring would be bicycles have gearing.
Winds and Hills will slow you down going up and into head winds, but work for you going down and with tail winds.
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Old 08-24-10, 05:03 AM   #10
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Since it sounds like your climbing is concentrated in the first couple days or so, sou can adjust the pace for the rest of the tour based on how that goes. If you are ahead after the first few days either finish early, take it easier for the remainder, or make an extra stop and do something else. If you are behind, either pick up the pace or cut the trip short at the end.

If at all possible flexible schedules without a set finish date are best IMO. Being a slave to a schedule sucks.

It never occurred to me that it was possible to really calculate what daily mileage was possible or more importantly desirable at least in the mountains. As a result I try to never really plan more than absolutely necessary. That means that I might say, "I either need to stop today at point a, or make it to point b due to the conditions ahead", but never try to plan where I will be more than a couple days ahead.

Truth be told when I do guess what I think I can do on a given day I am often way off in either direction. I have done 142 miles on a day I thought I might do 70 and have done 25 on a day I thought I would do 60.

There are places where lack of water, lack of places to camp and so on will mean that you will have to make a destination. Also in hot climates stopping in a place that allows the hardest climbs to be done in the cool of the morning is a good idea. All of this means some short term planning, but I try hard to only plan as far ahead as absolutely necessary and if possible to have more than one option for the day.

When it is flat as has been said the wind can be a factor and can also throw you way off. If you are not alone drafting can minimize the effect of winds though.

Bottom line for me is to try not to have a set schedule and to take it as it comes, planning only as much as conditions demand. It isn't always possible, but that is the best way to go if you can, at least it is to me.
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Old 08-24-10, 05:26 AM   #11
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Don't forget that not all hills are equal. If they're 2-5% grade, that's one thing, but if they're 6-10%, that's another when it comes to the time needed to climb them.

If you have the time, you might run down a copy of Performance Cycling by Stuart Baird. He uses some math and scientific principles to explain a lot about what happens when cycling, and although it is geared towards racing, the principles are the same. He has a section called "Nonuniform Conditions" where he talks about the effects of wind and hills and what strategies are best when encountering them. Although it sounds kind of geeky, it's actually written pretty well and easy to understand. I have recommended the book to others and they liked it.
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Old 08-24-10, 06:42 PM   #12
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Thanks, y'all. This helps!
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Old 08-24-10, 06:54 PM   #13
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I'm more inclined to have the math going into the exchange rate, dollars to Zloty I won, Dollars to Pounds I lost

But after a while it was too much work food = food, and beer =
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Old 08-24-10, 07:31 PM   #14
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Honestly I don't think hills matter in what distance you can cover in a day. If you want to go far a particular day which happens to have hills you just wake up a little earlier and pedal a little later. My longest day in both kilometers and hours of my 9 month tour up South America and across Canada was a day where I climbed three mountain passes. I knew the day was going to be hilly and I knew how far I wanted to get so I started at sun rise and went until just after sunset.
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Old 08-24-10, 07:51 PM   #15
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i wouldnt worry about it unless your heading west and gonna go over the rockies than at that point just double any numbers you have, cause guaranteed to have head winds no matter what direction your going and climbs that never seem to end
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Old 08-25-10, 04:58 AM   #16
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i wouldnt worry about it unless your heading west and gonna go over the rockies than at that point just double any numbers you have, cause guaranteed to have head winds no matter what direction your going and climbs that never seem to end
Don't discount the Appalachians. On the Trans America the Rockies were not particularly tough compared to the climbs in the East, despite the fact that the TA crosses the continental divide 9 times. The grades are long and not all that steep. In my opinion Virginia had the hardest riding of the whole trip and the Ozarks had some challenging climbs as well. I read somewhere that the Virginia part of the route even has the most elevation change of any state on the TA.

I have since found that not all western climbs are not of similarly moderate steepness though. On this year's tour I found the Sierras in SoCal plenty challenging.
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Old 08-25-10, 05:05 AM   #17
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The difference between hiking and bicycle touring would be bicycles have gearing.
Stride length = gearing.
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Old 08-25-10, 07:46 AM   #18
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Don't discount the Appalachians. On the Trans America the Rockies were not particularly tough compared to the climbs in the East, despite the fact that the TA crosses the continental divide 9 times. The grades are long and not all that steep. In my opinion Virginia had the hardest riding of the whole trip and the Ozarks had some challenging climbs as well. I read somewhere that the Virginia part of the route even has the most elevation change of any state on the TA.

I have since found that not all western climbs are not of similarly moderate steepness though. On this year's tour I found the Sierras in SoCal plenty challenging.
I grew up on in the east but have not had the chance to ride there so yeah my reply was kinda ignorant i just got back from a ride in the mountians and so im kinda frustrated my calculations were very wrong for time estimates.
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Old 08-25-10, 07:48 AM   #19
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For unladen biking (not touring) around town and for trips of two or three hours:

- Horizontal - I average about 13 mph on flat horizontal unpaved trails, about 14 mph on pavement.

- Vertical - On a grade of about 10 percent when I am in my lowest gear, I am traveling about 3.7 mph. On this steep a hill, almost all energy is applied to elevation gain, there is very little mechanical or rolling resistance and very little aerodynamic drag. Thus at a 10 percent grade that is climbing at roughly 0.37 mph in the vertical dimension which is equivalant to roughly 2,000 feet per hour. But at the top of the hill I am a bit winded so if I was planning for a long day with a lot of climbing I would assume that I could climb roughly 1,500 feet per hour instead of 2,000 feet per hour.

- Estimate times for horizontal and vertical, then take the sum.

Loaded touring, I would add 15 percent more time than I got with the above. This accounts for going slower with the additional weight, riding a bit slower with a loaded bike because it does not feel as stable, and pacing myself to go a bit slower so I can last all day.

You might want to calculate your own horizontal and vertical averages instead of using mine.
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Old 08-25-10, 08:52 AM   #20
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For unladen biking (not touring) around town and for trips of two or three hours:

- Horizontal - I average about 13 mph on flat horizontal unpaved trails, about 14 mph on pavement.

- Vertical - On a grade of about 10 percent when I am in my lowest gear, I am traveling about 3.7 mph. On this steep a hill, almost all energy is applied to elevation gain, there is very little mechanical or rolling resistance and very little aerodynamic drag. Thus at a 10 percent grade that is climbing at roughly 0.37 mph in the vertical dimension which is equivalant to roughly 2,000 feet per hour. But at the top of the hill I am a bit winded so if I was planning for a long day with a lot of climbing I would assume that I could climb roughly 1,500 feet per hour instead of 2,000 feet per hour.

- Estimate times for horizontal and vertical, then take the sum.

Loaded touring, I would add 15 percent more time than I got with the above. This accounts for going slower with the additional weight, riding a bit slower with a loaded bike because it does not feel as stable, and pacing myself to go a bit slower so I can last all day.

You might want to calculate your own horizontal and vertical averages instead of using mine.
That actually sound about right
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Old 08-25-10, 09:22 AM   #21
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Stride length = gearing.
Going up hills and even on the flats, that can make some sense, but hiking downhill and cycling downhill are much different. The other difference is that the hiker is always carrying the weight, so taking small steps doesn't have the same effect of diminishing the burden as using a small gear on a bike.
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Old 08-28-10, 09:59 AM   #22
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Dorky, I know.
Just wait, I'm about to make you look like the coolest kid in school, in comparison!

In general, I agree with the others who said that all the other variables that affect speed (wind resistance, type of hills, energy level, scenery, etc.) mean that predicting speed/distance based only one variable is rather futile. The first two, wind resistance and hill types, are the ones that can largely be ignored for hiking, but are important factors in cycling; on a bike, you can cover rolling hills pretty close to flat-land speeds because you don't use energy on the downhills and can store it for the uphills, but with hiking, you use energy both up and down.

However, the data hound in me couldn't resist doing some statistical analysis. If I plotted a large number of data points based only on daily climbing and average speed, would the the other variables be washed away in the noise, allowing a trendline to appear?

And, much to my surprise, a very distinct trendline *did* appear!



This is data from two separate tours, covering about 60 days of riding. The X-axis is that day's cumulative climbing (as reported by my altimeter-equipped cyclocomputer) divided by that day's miles. So, sort of an "average grade", that ignores all the downhills. The Y-axis is then just my total average speed for that day.

Google Spreadsheets doesn't show the trendline, but an equation OpenOffice came up with is:

AVERAGE_SPEED = -2.7 ln(CLIMBING) + 22.8

where CLIMBING means feet-per-mile.

My flat-land, no wind cruising speed is around 17mph, so perhaps a more general form of this equation would be:

AVERAGE_SPEED = -2.7 ln(CLIMBING) + NORMAL_CRUISING_SPEED + 5.8

Please use the term NDSL (Neil's Dorky Speed Law) when popularizing this concept!

Neil

Last edited by Neil G.; 08-28-10 at 10:08 AM.
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