Parallel question to this one: How does a cyclist's power output vary as gradient changes (assuming constant effort), compared to a baseline of cycling flat?

I'll state some of the obvious things:

• cycling fast downhill it's easy to run out of gears and eventually spinning the legs around doesn't actually produce any power
• cycling up too steep a hill means you cannot keep on pedalling, and so I guess power output goes to 0

But beyond that, how else does it vary? Can I expect that cycling for 1 hr at 5-10% gradient, I can average the same power output as for 1 hr on a flat road - clearly, at lower speed! The linked question mentions, correctly I think, that different muscle groups are used in climbing vs. flat routes, so it's possible to have one set well trained but not the other. This paper claims power output peaks at gradients around 6-7% regardless of cyclist's specialization (sprinter vs. climber).

It is a very open-ended question, but I'm interested in all aspects of how power output depends on the road gradient.

The background to this question is that I am a flatland cyclist planning a hilly cycling holiday, and I'm making some guesstimates as to the progress I can make on climbs.

• How is "constant effort" different than "constant power?" Merriam-Webster: "The meaning of EFFORT is conscious exertion of power." If you're making a constant effort, you're putting out constant power, and it doesn't matter what the terrain is doing. Jan 4 at 15:18
• @DavidW we should take power here in physics terms - something you can measure in W, for example by a crank power meter. Dictionaries are rubbish at physics, which doesn't conventionally define effort. So effort isn't measurable except subjectively. But if we imagine an really un-ergonomic setup, we'll feel like we're working harder for the same measurable power output. Or we'll actually be working harder, but not getting any more into the cranks. Can that efficiency vary? Yes. Standing and rocking the bike increases effort faster than useful power Jan 4 at 15:32
• There are clear and systematic differences in how one produces power on the flat and on hills. Whether those production differences are sustainable for equal durations is a tough question. Jan 5 at 3:09
• Do you have access to any kind of rideable hill anywhere near your home? If no, what about a bridge? Go there and ride it 10 times, and see how you feel during and afterward. Use Strava or similar if you can to get data.
– Criggie
Jan 5 at 7:00
• Can we assume an unlimited gear range for simplicity’s sake? Jan 5 at 10:31

There are two physiological facts that need to be considered here:

1. Muscles consume energy not only to convert it into mechanical energy, but also to sustain a certain force. The greater the sustained force, the more energy is wasted into just sustaining the force.

2. Muscles cannot work indefinitely fast. There are two broad categories of muscle fibers which are distinguished by how fast they can twitch. The important fibers for biking are the slow twitch fibers because they burn their fuel with oxygen. The fast twitch fibers generally like to use anaerobic chemistry, and are thus not suited for endurance sports like biking. When the speed of the muscle contraction gets into vicinity of the fibers twitch speed, force output drops overproportionally, making the muscles inefficient.

As such, the perfect power output is achieved with a cadence that is as high as possible (less force = lower losses due to effect 1) without loosing too much force due to effect 2. That is the reason why professional athletes like to spin at around 90rpm.

Now, getting back to your biking: If you know your power output on the flat and the lowest speed at which you can maintain your cadence by switching gears, you can derive the grade on which you can still ride that speed. As an example, assume `200 W` power output, min speed of `10 km/h`, and a cyclist/bike weight of `100 kg`. This rider can sustain a vertical speed of `200 W / 1 kN = 0.2 m/s`, which translates to a grade of `0.2 m/s / (10 km/h) = 7.2%`. This will be close to the most energy efficient climbing speed, and you will be able to endure that as long as you can ride on the flats. Any stronger ascent will reduce your climbing performance up to the point where you have to give up and give in to pushing your bike uphill.