Does anybody know of a good formula that predicts how running speed changes as a function of the slope? With this info and some topographic maps it would be possible to compare times on courses of varying severity.
Speed vs. Slope
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Good question. I've always wondered how much I lose by going up a hill relative to how much I gain while going back down. It would be neat to calculate youe expected splits whijle taking toppograpy into consideration. I suspect it varies too much runner to runner because of form or even a tendency to work the hills or not to come up with a decent formula

My coach has always said it takes about 40% more energy to maintain the same speed on an incline. That is a pretty generic guess since that doesn't really explain the grade of the hill, but that is my insight.

Interesting question ... I've been training for some uphill mountain races. The group that I have been running with has come to the conclusion that we should judge our "pace" by rate of climb ... feet/minute, feet/hour rather than min/mile miles/hour. Obviously pace per mile is significantly different for a flat 10 mile run vs a 10 mile run that involves 3,000ft of climbing. Despite the fact that you might only be running at 9:00 min / mile ... you have to factor in the rate of ascent.
I think that you could determine the answer to the question with the aid of an HRM ... where you would run a measured distance on the flats, maintaining a set HR, then run the same distance on a known incline, for the same distance, while maintaining the same HR, ... then doing the same for other inclines. I would say that a treadmill would be good tools, but treadmills are notoriously incorrect with regard to incline and especially distance. 
This may help:
ACSM equations for VO2 used at a given velocity:
level ground  VO2 = (velocity x 0.2) +3.5
When running up a grade, the additional O2 consumption that must be added to the horizontal component to calculate the vertical component is:
VO2 = velocity x grade x 1.8 (overground running) OR
velocity x grade x 0.9 (treadmill running)
velocity  m/min
VO2  ml/kg/min
grade  expressed as a decimal (eg. 2% = 0.02)
By having both equations' VO2 values equal, you can ensure an "evenpaced" run, despite the effects of the incline. 
Something seems wrong with those equations. For a level ground running velocity of 300m/min, VO2=63.5
VO2 = (300 x 0.2) + 3.5 = 63.5
That sounds reasonable for a VO2 at that speed. However, when you go to the graded VO2 equation, things get messy.
VO2 = velocity x grade x 1.8 (overground running)
If we use the same velocity of 300m/min and a grade of 2%, we get:
VO2 = 300 x 0.02 x 1.8 = 10.8
In order to have the same VO2 of 63.5 while running up a 2% grade:
VO2 = velocity x grade x 1.8
VO2/1.8 = velocity x grade
63.5/1.8 = velocity x 0.02
velocity = 1763.9m/min
This is obviously WAY too fast
If we're calculating a relative VO2, then body weight must be taken into account; if an absolute VO2, then it won't be needed.
If you have a link to those equations post it and I'll try to fiddle with them some. 
sorry for the confusion. Let's see if I can clarify what I was trying to talk about...
The VO2 equation for running up a slope must be added to that of level ground VO2. Therefore, VO2 for running up a 2% grade at 300m/min = 63.5 + 10.8 = 74.3 ml/kg/min.
Now, for "even paced running" when running up a slope, the 2 VO2 equations must be equal:
let level running velocity = y and slope running velocity = z
(y x 0.2) + 3.5 = [(z x 0.2) + 3.5] + [z x grade x 1.8]
For a grade of 2% and a level ground running speed of 300m/min, the velocity when running up the hill must equal:
0.2y+3.5 = 0.2z+3.5+0.036z
0.2(300) = 0.236z
60 = 0.236z
z = 254.24 m/min
Therefore, to maintain "an even pace" when running up a 2% grade, you must slow down from 300 m/min (level ground) to a velocity of 254.24 m/min while running up the slope.
Hope this makes sense.
fletch 
So this would imply that
y/z = 1 + 9*slope,
where y = running speed, z = slope running speed, and the slope is in decimal notation. What is interesting about this equation is that it's independent of the runner.
This can be used to provide a "difficulty" rating for a race course, given its topographical map. Thanks for the info! 
I'm not exactly sure how you could calculate a "difficulty" rating for a course unless either:
1) the course was all flat and/or uphill
or
2) we had a formula for the effect downhills have on VO2
I'm sure the uphills would slow a runner much more than the downhills would help a runner. I really like your idea of developing a "grading" system for courses. Maybe we'll come across a formula for VO2 on downhills somewhere.
These formulas can be used relatively well to calculate splits for a race if we're only calculating flat/uphill portions.
Example: Splits for an "even effort" through the first 4 miles of a marathon, where the course is:
2 miles flat
1/2mile uphill at 3%
3/4mile flat
1/4mile uphill at 5%
1/4mile uphill at 2%
1/4mile flat
Assuming a velocity of 300m/min on the flat portions, his/her corresponding velocities on the hills would be:
2% = 254.23m/min
3% = 236.2m/min
5% = 206.9m/min
These would have to be converted from metric to English units in order to get mile splits. I'm not going to type all the calculations, but in order to maintain an "even effort" (VO2 of 63.5) the runner would need to run splits of:
Mile 1 = 5:21.9 ~ 5:21.9
Mile 2 = 5:21.9 ~ 10:43.8
Mile 3 = 6:05.3 ~ 16:49.1
Mile 4 = 6:12.6 ~ 23:01.7 
By the way, I found the uphill velocities a little quicker by solving the equation for the uphill velocity.
x = flat running velocity
y = uphill running velocity
g = grade
y = 0.1x/(0.9g + 0.1) 
CarolinaRunner, you make a good point; on further analysis, this formula suggests that when one races down a hill with a grade of 11% or so, then the speed becomes infinite  hardly realistic! I agree that the uphill formula is probably valid.
Perhaps this can be patched up with further assumptions. Thanks for your input. 
GLT,
Maybe it's true that as the downhill approaches 11%, speed at a given rate of oxygen consumption COULD approach infinity, less other unaccounted for elements such as wind resistance. At such a steep decline, the runner's would no longer be limited by oxygen consumption, and would be able to run as fast as his/her legs could move. Therefore, the downhill equation would need to approach some constant which is the maximum attainable speed for the runner.
However, at such speeds, wind resistance and resistance from the friction of the road, shoes, etc, would contribute to oxygen requirements as well as the maximum attainable downhill speed.
I know nothing about physics, but I'm guessing gravity would also be in the equation somewhere. 
I figure that every 1% is equal to about 12 seconds per mile faster. So, if you are running on a slope that is 2%, your energy expenditure is equivalent to running 24 seconds per mile faster, for example. If you run on a treadmill and speed up by a minute per mile and record your heart rate, then go back to your original speed and increase the grade by 5% your will find your heart rate equal to the minute per mile faster, approximately.

I found a passage in Tim Noakes book about uphill and downhill running. He states on page 61 (newest Lore of Running book)that C.T. Mervyn Davies studied energy expenditures and found that every 1% of uphill running required 2.6 ml of O2 more, equivalent to .65 km per hour reduction in running speed. Addtionally, downhill running required 1.5 ml of O2 less for every 1% change or an equivalent speeding up equivalent of .35 km per hour.
A side note, on the previous page he stated that research showed that anyone running slower than 2:21 marathon pace (about 5:24 pace, I think) does not benefit from drafting. Interesting! 
I calculated the effects of both up and downhills on Boston course, quite a few years ago and it was published in a little handout that went with the race packets. I have a copy of that somewhere. I also have formulas that make both up and downhill conversions in some programmable calculator somewhere around here. Seems that each % up hill slows you about 15 sec per mile and each % down gives you about 8 seconds per mile benefit, provided that you maintain the same energy expenditure. Another calculation I had showed about 12 sec per mile lost per % up. I've been tryng to get some runners to do another study on this. It isn't hard to do  just run some repeated 5min runs at different grades and calculate how big an increase you get in VO2 with each % grade. When we did this before I remember that different people respond differently  some handle hills better than others (who may handle speed increases better than the hill people). A problem is that the up grade increases the cost so much that it is hard to run very fast, because the VO2 will go above max real quickly. So yu end up extrapolating from slower speeds and hope it applies at faster ones. I have done faster ones using Rate of Perceived Exertion and that can be done beyond max, but not ver exact

JT,
Interesting that you wrote about a 1215 seconds difference in pace for every 1% uphill. I wrote 12 seconds on my response above yours. In that response, I had written 15 seconds because of my personal obersvations, but I changed it to 12 seconds to be conservative. I recognized on a treadmill long ago that at pace slower than 9 mph I had a change in pace equivalency by about 15 seconds per mile, but at speeds faster than 9 mph I had 12second equivalencies, especially in the 1012 mph range. Tom S. 
trail runner wrote:
A side note, on the previous page he stated that research showed that anyone running slower than 2:21 marathon pace (about 5:24 pace, I think) does not benefit from drafting. Interesting!
I'm sure there would be some benefit in a strong headwind. 
According to Davies research, no advantage is gained by drafting until a runner reaches 18km per hour or 5:24 pace. Interesting, isn't it?

When hiking there is a general rule of thumb that i have heard (and has seemed to work fairly well in practice), that you add 1 mile of time for roughly every 1000 feet of elevation increase. Maybe this translates to running as well? Maybe a more precise number exists?

I think that rule is probably alright for the average, nonexerciser, when they hike, but it sure is far off for wellconditioned persons. Part of your hiking equation depends upon the terrain too. If it is rocky and poor footing, then slowing down occurs a lot, especially for nonfit persons. I lived in Colorado and hike up Long's Peak a few times. Most people left at 4 AM to reach the top by noon and then returned by 8 pm. My exgirlfriend and I would leave at 8 AM or later and be there by noon, stopping to eat and take pictures along the way of animals. We didn't hurry by our standards and were up there, rested for half an hour, looking at the scenery, and back by 4 pm. So, it seems to me that our pace was not slowed very much. I bet we slowed only 1015% even though it was relatively steep in some sections and semisteep in others. To a degree the severity of performance decline is due to general conditioning and to a degree specific conditioning. A lighter person climbs faster too, just like in cycling lighter people have advantages. I suspect that light, but strong runners slow far less than heavier runners that are of equal ability on the flats.