# Estimating climbing speed on real-world climbs with variable grade

There are numerous calculators online which estimate climbing speed based on power and grade or similar. For example:

But real-world climbs are rarely precisely the same grade throughout. For example, Hawk Hill (a popular climb in the San Francisco area) averages 6% grade, but has sections as steep as 11% and even a slightly downhill section.

Of course, aerodynamics, rolling resistance, and drivechain losses will complicate the picture, but let's limit the issue to climbs which remain slow, steep, and sustained enough that still overcoming gravity is by a wide margin the most significant force for the rider to overcome, and thus aerodynamic drag and rolling resistance don't vary significantly over the climb.

In such circumstances, is average grade sufficient to relate power and climbing times, as exemplified in the calculators above? Or does the variability in grade introduce significant error? How significant? Please show with math, if possible.

For the same average gradient and same constant power output, and assuming same bike, same tires, and zero wind the variable grade climb will take longer than a constant grade climb.

The reason is that with constant power and a variable gradient, you go faster when it's shallower (and slower when it's steeper) but aerodynamic drag increases with the square of air speed so you don't gain as much speed on the flatter parts as you lose on the steeper parts.

Suppose you were comparing two climbing routes: route A is a constant 5% over 3 km; route B is flat for 1.5 km, then climbs at 10% for 1.5 km. Both have a total length of 3 km, and both climb a total of 150 meters. Ignore for the moment the difference between road distance and horizontal distance.

At a constant 250 watts, the same CdA = 0.3 m^2, the same Crr = 0.005, rho = 1.2 kg/m^3, and total rider+bike mass of 80kg, the average speed for route A is 5.21 m/s, or a total time of 576 seconds.

For route B, the speed along the flat section is 10.51 m/s and the speed along the 10% gradient section is 2.98 m/s, for a total time of 646 seconds.

As an aside since you did not ask the question, constant power on route B is not the fastest way to do this climb. Constant power is time-minimizing only when the conditions are also constant so a time-minimizing strategy for variable gradient (or variable wind, or variable surfaces) is to vary the power. There are physiological constraints, of course, on how much you can vary the power so the optimization problem can be complex.

• I think you may have missed this part of the question: "let's limit the issue to climbs which remain slow, steep, and sustained enough that still overcoming gravity is by a wide margin the most significant force for the rider to overcome." A "climb" where half the climb is flat is not that. – Phil Frost Feb 14 at 0:05
• The math is similar if the segments were 4% for 1500m and 6% for 1500m: the variable climb is slower than 3km @ 5%. The reason is exactly the same even if the example gradients are different. – R. Chung Feb 14 at 0:52
• I really like how you bring accurate numbers to these kinds of answers, and explain how they apply. I learn something pretty-much every single time. – Criggie Feb 14 at 4:13
• The question supposes that the differences in aerodynamic drag and rolling resistance are negligible. In such a case, does variable grade matter? – Phil Frost 2 days ago
• Hmmm. Well, Crr is independent of slope so there's no difference in rolling resistance at all. However, as long as you're traveling within the atmosphere, aero drag will differ with speed. If you were traveling in a vacuum on the surface of the moon, there is no aero drag and speed would be proportional (linear with) gradient. In that case, variable grade does not matter and you could use average grade. On the surface of this planet, variable grade is slower than constant grade. – R. Chung 2 days ago

If the grade did not matter, then you'd be able to maintain the same VAM on any grade. For climbs like you describe VAM is going to be the major determinant of overall time on the climb. For most riders their VAM vs grade forms a bell curve. Based on your fitness and gearing, there will be a grade on which you can maintain your maximum VAM. For shallow grades, you simply can't ride fast enough to maintain the cooresponding vertical velocity and for steeper grades you are limited by the gearing to cadences that limit power output.

The wider your gear range the steeper grade you'll be able to sustain your maximum VAM. However, eventually even the lowest gear will require more power than you have to keep the same VAM.

For an interesting application of this see this article on the 48 hour climbing record. The rider chose a very specific section of climb that had a gradient on which he could record his personal maximum VAM. I've ridden the entire climb many times, I can't imagine just doing the steep part for 48 hours.

Now as far as the meat of your question goes, I'd say it's depends on how you're using the calculator. If your using them to get some estimate of your avg power on a climb, then using the average gradient is an acceptable estimate of the total work. If you are using them to estimate your time on a climb, then the inaccuracy of the input data far outweighs any variance due to gradient change.

• "then the inaccuracy of the input data far outweighs any variance due to gradient change" in accuracy of what input data exactly? Why would the calculations be valid in one direction (calculate time by average power) but not in the other (calculate average power by time)? – Phil Frost 2 days ago
• @PhilFrost Actually, if you know gradient, CdA, Crr, the air density rho, and the total mass then, yes, you can accurately calculate both average power if given speed, and speed if given power. The exact equation is well-understood if not always well-known. You can find a discussion of it here in this bikes.SE question and answer. – R. Chung 2 days ago
• It is one thing to put an power output into a form, it is completely another to maintain that power output steady for an entire climb. Is your water bottle full? What is your exact weight at the bottom of the climb? What does your bike really weigh? – Fred the Magic Wonder Dog yesterday