# Power and climbing speed

I live somewhere very, very flat (bridges are the biggest hills around) and I'm planning some hilly routes for a bike holiday. I have a decent idea of how much power I can produce, but less sure how that converts to distance covered on a hilly ride - and can't easily test locally.

Simple physics suggests that given some power P, mass (me+bike) m and gravitational acceleration g, climbing speed in m/sec would be P/mg, or more usefully, 3600P/mg meters per hour. So for 200W and 85kg me+bike, that'd be 860m/hr climbing distance.

This ignores air drag and bike inefficiency, which I guess are minor (at 5% incline that's only 17kph horizontal speed, and half that at 10%) - but otherwise, is that roughly right? Can I conclude that if I'm happy putting down 200W for an hour, then I can climb roughly 800m in that time?

This won't lead to any life or death planning questions, but equally I want to know where to aim, so I can narrow down some route options.

EDIT: to clarify, I'm also interested in how my power assumptions translate from flats to hills. I will still be cycling at low altitudes so that's not an issue, but climbing is of course not the same as cruising on a flat road...

• Off the cuff intuition: my riding group and I are in the about the power/weight class as you: I’d plan on about 24 km/hour with a golden ratio course of about 500 m of climbing per 25 km of riding. Jan 1 at 15:22
• @PaulH What's a "golden ratio course?" 500m of elevation over 25km might or might not be climbing depending on the course; a steady 2% grade isn't really "climbing." Jan 1 at 16:29
• @DavidW In the PNW, people talk about the golden ratio for a road ride, where at the end of the ride you’ve climbed about 1000 ft for every 10 miles. Sure, that can come in a lot of forms (slow and steady vs flats punctuated by steeps). But for planning on timing, nutrition, and hydration, it tends to work out. Mountain bike rides west of a Cascades, the typical ratio works out to about 2000 ft - 2500 ft of climbing for every 10 miles of riding. It’s not a perfect model, but it’s a useful one. Jan 1 at 16:34
• If you have not trained on hills, it would be naive to plan to do a steep climb at the same sustained power as you ride on the flat. Slightly different muscle activation at slower speeds and changed bike angle, along with the phycology of putting in that much effort and going that slow will have an impact. You might make the transition to being a hill climber seamlessly, but you might not. Jan 1 at 22:19
• My gut feeling says that 800 meters per hour are perhaps slightly optimistic, especially when you're no mountain goat by nature (or at least not yet). Only for reference: I did a competitive climb up a mountain pass in summer and managed about 850-900 meters of ascent per hour (for 2 hours), at avg. 200W but I weigh in about 20 kilograms less and I knew I only need to ride to make it to the finish line. I would also factor in gearing, at least on prolongue climbs, for most it will be more fatiguing grinding away at low cadence, so 53/39 - 11-25 might not be your choice for mountain passes. Jan 2 at 16:53

Here is a calculator where you can play around with these numbers. At 85 kg rider+bike and 200 W output, you'd be moving 14.4 km/h on a 5% grade (assuming no wind), and 8 km/h on a 10% grade.

You are correct that at these low speeds, aerodynamic drag is small; drivetrain and rolling-resistance losses should also be low, although that will depend on your bike's tune, your tires, and the road's condition.

The answer to your "is that roughly right?" question is: yes, roughly.

You're essentially asking for a relationship between watts/kg and VAM (velocità ascensionale media, a term coined by Michele Ferrari the famous (or infamous) Italian physician and cycling coach). VAM is the gain in elevation (in meters) one could achieve in an hour for a given power and constant slope. Ferrari noted that a pro rider producing about 6 watts/kg would have a VAM of around 1600.

Here is a graphic that shows how many meters a rider producing 175, 200, and 225 watts with a total mass (rider + bike) of 85 kg could expect to climb in one hour at various gradients. This graphic takes into account rolling and aerodynamic resistance, but at steep slopes the amount of power allocated to these are small and the amount of power allocated to climbing dominates.

Although VAM doesn't have a true asymptote, you can see from the graph a key characteristic of VAM: that as the slope steepens, the change in VAM decreases. This is the basis for the approximation that "maximal" VAM doesn't depend very much on the slope: once the slope becomes steep, VAM doesn't change much.

Not obvious in this graph, but often helpful to know, at the far right edge (i.e., for steep slopes), the difference in VAM between the various powers is almost proportional to the difference in power: that is, a x% difference in power will translate to almost exactly a x% difference in VAM. That's because at steep slopes your speed is low and when your speed is low the aerodynamic component of power (which varies with the cube of speed) becomes less important and is dominated by the power used to climb (which varies linearly with speed). The slower you go, the more closely power and speed are linearly related. Thus, if your target were 200 watts but you could only average 180, VAM would be very nearly 180/200 = 90% of the VAM at 200 watts. If you could produce 210 watts, VAM would be very nearly 5% more.

• I read the OP's statement of "85kg me+bike" to mean that the OP weighs 85kg and the bike will be an additional mass. Also where did that graph come from? Jan 1 at 23:48
• @PeterM I calculated the curve for 85 total kg, and average values for CdA, Crr, and rho; and presuming no wind. Jan 2 at 0:09
• Thanks, perfect! And... Does the power output depend on gradient? From a physics point of view it shouldn't, but as a matter of practicality, does it? I'm sure at steep enough gradients yes, but what about the range say 5-15%? Jan 2 at 8:34
• @Bennet: That’s actually a great question. Somehow when riding uphill it feels more natural to ride on the tops of road bike handlebars and this upright position makes the glutes and hamstrings work harder. Most people seem to have a higher power output when going uphill, but I’m not sure if this is psychological, physiological, a lack of gears or intentional tactics (uphill every watt gives you a 1:1 improvement in speed and time, on flat terrain you have diminishing returns due to aerodynamic drag). Jan 2 at 9:43
• @Bennet That's a complicated question, and you might consider asking it as a separate question rather than have it buried in an answer here. Jan 2 at 16:05

Your main problem here will be pacing a climb such that you can get up it without "popping" part way up.

Given you're not racing, then your goal should be simply to achieve the climb.

It's possible you'll be able to smash the first 5 to 10 minutes of the climb at the high power output, and then feel awful for the rest of it.

If you use Strava or similar, you could compare your time/s on any decent climb segment with another local rider who is of similar strength. Do remember that Strava only shows you their best-ever time, and it could be years ago, maybe in a race, and probably unloaded with a tailwind. Aim for 150% of their time for that whole segment.

If software's not your thing, then aim for an effort level where you can speak out-loud a whole sentence without having to pause to puff in an extra breath. The upper limit should be a three-word-phrase. Don't go so hard that you can't say more than a single word at a time. Eg

• "If you can say a whole sentence at once" Target pacing
• "If you can say a, whole sentence, at once" Upper level of effort, perhaps for a ramp or a corner or a brief steep bit.
• "Ifyou, cansay, a, whole, sentence, at, once" Too much - dial it back
• "If." Way too much! Relax and perhaps have a breather on the next safe spot.
• I think pacing is clear - or at least, that's why I'm looking at power. But what isn't clear is what pace I can assume at these effort levels. Jan 2 at 8:15