Here is a list of the responses I have received for my question:

Hello all- When runners reach VO2max in the laboratory they are often (always) running at some incline. However, runners like to get a velocity associated with VO2max to help pace themselves during workouts. Are there any calculations that relate this velocity to the velocity they would be reaching on a flat?

Maybe Power = Force * Velocity. I just don't know what the force would be. mg? resistive forces? GRF? Or maybe I am way off base and calculations already exist?

Any help or insight would be greatly appreciated.


1. You are way off base with Force*Velocity if velocity is the speed of the treadmill and force is related to the weight vector. The power of a force is the dot product of the force and the velocity which implies that the force and velocity must be parallel to each other. In the case you described the force of gravity (or a component of it based in the treadmill incline) is perpendicular to the velocity. A possible solution is to use the equation Power= external work / time. The external work would be the change in height due to running up the incline. Of course, this would mean running level does no work and running downhill would produce negative power. This method assumes the running velocity is constant and that internal work is not to be included (internal work is the work done to cycle the limbs each stride. Thus your power is:

power = [velocity * sin (angle of incline) * mass * 9.81) ]

This equation looks odd because the is no "time" variable but time disappears because it is in the numerator and denominator. The work done increases with the amount of time and the power decreases with the time, i.e., they cancel. The mass can also be eliminated if you want a power normalized to body mass so that heavier runners aren't given an advantage. Of course, there is no real advantage because they have to carry their own mass up the incline. The 9.81 is acceleration due to gravity. It can be eliminated if you use body weight as your normalization factor, i.e., watts/newton vs watts/kilogram.

I'm not a researcher, just a layman interested in biomechanics for his own purpose. I don't think the VO2 max test on a treadmill is directly applicable to running on a track. Running efficiencies are slightly different, especially if there is a pronounced incline. Also a 1% incline is used in treadmill testing to compensate for the lack of wind drag in running. But if you want to know the velocity you can go to Jack Daniel's (or any other) VO2 max predictor, type in the VO2 max and get the velocity that corresponds. Or I saw this:
The standard predictions for calculating VO2 Max are:

percent_max = 0.8 + 0.1894393 * e^(-0.012778 * time) + 0.2989558 * e^(-0.1932605 * time)
vo2 = -4.60 + 0.182258 * velocity + 0.000104 * velocity^2
vo2max = vo2 / percent_max

where time is in minutes and velocity is in meters per minute. These equations are also used for working backward to determine a time corresponding to a known VO2 Max and distance, although it requires approximating percent_max, combining equations, and treating vo2 as a quadratic equation to solve for velocity, which is in turn used to calculate time (time = distance / velocity) and check how close the initial time estimate was.

I have no idea if this helps since it doesn't address the issue of changing an inclined measurement into a straight one.

3. Check out the research on "running economy." You can
test oxygen consumption at several submaximal speeds
and extrapolates to a known VO2max value to estimate
speed at VO2max. The ACSM's running equation can be
used to get a rough estimate of speed at VO2max or to
find the submaximal speeds to use. It is...
VO2 = 0.2(speed) + 0.9(speed)(fractional grade) + 3.5
VO2 is in mL/kg/min. Speed is in m/min. Hope this
Meredith Olson

4. I would guess the best method would be to use the VO2 equations for
Resting, Vertical and Horizontal oxygen cost.
VO2 in (ml / kg x min) = V + H + R (as above)

which =3.5 + (0.2 x speed - in m/min) + (0.9 x speed - in m/min x slope - in
i.e. 20% slope = 0.2

assuming vo2 consumption at max effort is the same whether running on an
incline or a flat... you can solve for the speed with 0 slope.

Also, I am guessing that these runners were on a treadmill. In the
running world, a slope on the treadmill of 1-2% is considered more
representative of actual running.

I hope this helps.


5. You can find equations that determine VO2 from speed and grade for running
in ACSM's guidelines for exercise testing and presription (I don't have my
copy handy or I'd send you the equation). I think you could input VO2max
and determine speed using 0 incline.


6. first, i don't see why you could not reach VO2max during level
treadmill running. Jack Daniel's has a flat VO2max protocol i think.

but anyhow, i would think that a useful test for runners would be of
their economy during flat running at say 3 or 4 speeds.
then, i'd apply a linear best fit line to those data and extrapolate
out to vo2max, drop down to the velocity axis and there you go,

using any sort of equation would be an average value and thus miss
the nuances of an individual test.

we can talk on tuesday if that didn't make any sense.


7. Maybe some measures of submaximal running economy on flat would give an expression of the increase in VO2 with increased velocity. Only (major) problem is, that running economy probably decreases when the intensity becomes close to maximal...

Best regards
Jesper Bencke

Institute of Exercise and Sports Sciences,
University of Copenhagen

THANK YOU ALL for the insight. I understand the utility of using submax relationships between speed vs. oxygen cost to extrapolate a maximal value, however, I am convinced that this relationship deviates from linearity. I think, after reading all of your responses, that using the ACSM equations, setting the fractional grade to zero, and solving for velocity is the way to go.

Thank you again-

Todd Carver

Biomechanics Laboratory
Boulder Center for Sports Medicine
(303) 441-2220

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