If Strava tells me that this month I have biked about 500 miles with 35,000 ft of elevation gain, about how many miles would that be equivalent to if the elevation gain were a few hundred feet?

I think it's still 500 miles.

500 miles.

If you would ride 500 miles
and you would ride 500 more.....

What is 500 miles, Alex?

Holding what constant? Total energy? If so, yeah, more than 500, but there is no universally valid conversion.

BF will give you credit for 507 miles since you did go up about 7 miles. Nicely done.

Ask Mister Crow

Where did you start and finish? If you start and finish in the same place for every single foot of that 35,000 ' elevation gain you also had a foot of downhill coasting.

35,000' in 500 miles is about 70' per mile. That is about 1.3% grade.
It would be slightly, very slightly less than the 7 miles CAT7 credited you with since it would be the hypotenuse of your imaginary triangle, 500 miles along the base and 500' high

It can be calculated, but it's complicated and depends on a lot of variables. Most importantly, we'd need to know:
1. Disc or rim brake?
2. Mechanical or electronic shifting?
3. Tubeless or normal clinchers?
4. 1x or 2x?
5. Steel or carbon?

So I think I have to put the question a little better than I did originally...

Person A rides a bike 500 miles, and does 35,000 feet of climbing, and, of course, does 35,000 feet of descending.
Person B rides a bike on the flat. No climbing or descending. Person B uses the same energy as person A.

How far, approximately, will person B have ridden?

If your answer is 500 miles, then, with all due respect, you haven't ridden mountains.

Climbing is probably a better measure of a cycling workout than mileage, except where mileage appears in the denominator, so you can get an idea for how steep the climbs are (greater than 100 ft/mile indicates you are doing more than filling in junk miles).

Neglecting aerodynamic savings due to reduced speed as this is extra math I don't feel like doing...

So at ~140 (Strava) watts I do about 17mph. So here I am at about 3600s*140(j/s)/17mi = 30 kJ/mile. I live in Illinois so I'll neglect my 10'/mile

at 70kg with 1100m of climbing, this is 70(kg)*11000(m)*9.81(n/kg) = 7,500 kJ

so 7500(kJ)/30(kJ/mi) = 250 extra miles probably +20/- 50% cause I have no idea how good Strava watts are, but it should get you in the ballpark.

Are you still at 35,000 ft?

John

Well, 70 feet per mile isn't exactly mountainous. So without doing any real effort at scientific accuracy, I'll estimate I'd ride a typical 70 ft / mile ride at around 15 mph avg (depending on so many other factors). And a perfectly flat ride at 21 mph avg (depending on so many other factors).

So the hilly ride would take me 33.3 hours. And in that same time I would ride 700 flat miles.

I know you're looking for more scientific than that based on power output. I'll argue that in 700 flat miles I would pull my hair out and give up before I finished....SO boring...never get to change pace or speed, never climb, never coast, never descend. Ugh.

In my case, my >100ft/mile rides average about 5 mph slower that my relatively flat rides. This is both because I get a lot slower when climbing steep hills, and I am a wuss on the descents. But if I use this as a crude estimate, 500 miles of up/down = 750 miles of flat.

Someone mentioned if you are riding in a closed loop, you have no net elevation gain and therefore, implicitly, nothing that differs from a flat ride. That isn't true, because a major component is overcoming gravitational potential energy (mgh, m = mass of rider+bike, g is the gravitational constant, and h is the vertical height climbed). It takes calories to surmount that potential energy barrier, but you don't regain those calories when you coast down the hill. Same thing with friction (tires on the road, wind resistance, etc).

With all due respect, your problem is ill posed. Unless you specify the length and grade of the climbs, there’s no way to answer. Also, normalizing by energy won’t work. For very steep climbs, energy will be nearly independent of speed, but as soon as gravity effects become small, the energy to cover a distance is highly speed dependent. In other words, there is no unique distance that can be covered for a given energy expenditure.

If you drop a pound of lead and a pound of feathers off a 100' building which will hit the ground first?

The lead since the feathers have more resistance to air.

BTW... the final climb in today's Vuelta Espana


Alto de L'Angliru
Gain in altitude: 1,266 m (4,154 ft)
Length of climb: 12.5 km (7.8 mi)
Average gradient: 10.1 %
Maximum gradient: 24 %

Not necessarily. It depends on how the lead and feathers are shaped.

True.. if the lead is say a sheet only 2 microns thick, it could flutter like a newspaper.

IIRC from high school, all physics problems are solved ignoring resistance due to air because it's too hard. In that case, then all you're doing with climbing/ascending is adding/subtracting equal amounts of work from different segments of the ride. Also ignoring human physiological considerations, as they are too hard. So the HS physics answer is 500 miles.

In the non-vacuum world you lose efficiency on descents due to wind resistance increasing as the square of speed. I have no idea how much though as it's very dependent on aerodynamics, wind direction and speed and I suppose even air density. Also I think the higher power output on climbs has some negative physiological impact, but I have no data telling me that's true, and it can anyway be mitigated by gear selection.

It's just the sort of problem that can be fun to attempt solving whilst riding 500 miles/35k feet. Get an estimated CdF, find the formula, an do math in your head for a few hundred miles. Makes me wonder, does doing math in your head whilst riding make you faster, or slower?

You recall wrong. Just because you can’t imagine something is possible, doesn’t make it so. From what I hear, aerodynamics is a fairly mature and robust area of study. I hear there have even been a few notable successes.

You won't see me going up in any of those newfangled flying machines. If God had meant for us to fly he would have given us wings.