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Any science nerd here work out the work equivalence formula for incline vs. flat ?

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Any science nerd here work out the work equivalence formula for incline vs. flat ?

Old 07-25-22, 09:55 AM
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Any science nerd here work out the work equivalence formula for incline vs. flat ?

For example, 1 minute up a 6% grade is equal to X minutes on a flat grade.
If not specifics, has anyone ever come up with a "Rule of 72" style rule of thumb approximation?
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Old 07-25-22, 10:11 AM
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Originally Posted by CheGiantForLife View Post
For example, 1 minute up a 6% grade is equal to X minutes on a flat grade.
If not specifics, has anyone ever come up with a "Rule of 72" style rule of thumb approximation?
Power to ride a bicycle can be calculated pretty accurately with a well known equation:horse power = [Vg*W(.0053 + %G/100) + .0083(Va^3)]/375

calories/hr = [Vg*W(.0053 + %G/100) + .0083(Va^3)]*7.2

watts = [Vg*W(.0053 + %G/100) + .0083(Va^3)]*2

where Vg is ground speed, Va is speed through the air (includes head/tail winds), W is bike + rider weight in lbs., and %G is grade in per cent. The factors listed here (0.0053 for friction + rolling resistance and 0.0083 for aerodynamic drag) are obviously not absolute. They will vary with efficiency of the tires and drive train, and with the aerodynamics of the bike + rider combination. Both of these assume a racing position on a racing bike. A clunker bike or a more efficient riding position will change these numbers, which are averages anyway. Power to overcome friction and gravity is proportional only to rider weight and ground speed. Power to overcome wind drag is proportional to the cube of the air speed. For reference, 1 hp = 2700 calories (because of human metabolic efficiency of 24%); 1 calorie = 0.276 watts; 1 hp = 746 watts. Here, all calories are kg-calories, or "food calories."

Analyticcycling.com allows you to put your numbers into a calculator and get numbers out. There are other calculators out there but some of them are pretty inaccurate. They often estimate significantly high calories burned.
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Old 07-25-22, 10:19 AM
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Originally Posted by CheGiantForLife View Post
For example, 1 minute up a 6% grade is equal to X minutes on a flat grade.
If not specifics, has anyone ever come up with a "Rule of 72" style rule of thumb approximation?
It would be easy enough to use a website calculator to get a table across a few representative grades.

Bicycle Speed (Velocity) And Power Calculator

Last edited by njkayaker; 07-25-22 at 02:06 PM.
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Old 07-25-22, 10:22 AM
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Originally Posted by CheGiantForLife View Post
For example, 1 minute up a 6% grade is equal to X minutes on a flat grade.
Sure, let me do the math for you:

1 minute of hard work up a 6% grade = 1 minute of hard work on a flat grade.
1 minute of easy effort up a 6% grade = 1 minute of easy effort on a flat grade.
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Old 07-25-22, 10:27 AM
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Depends if bare chested or if the dumpster dove smirnoff was consumed. This would require nonlinear approximations.
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Old 07-25-22, 10:35 AM
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Ask this sort of question in the Nutrition and Training forum.
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Old 07-25-22, 10:39 AM
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Originally Posted by WhyFi View Post
Sure, let me do the math for you:

1 minute of hard work up a 6% grade = 1 minute of hard work on a flat grade.
1 minute of easy effort up a 6% grade = 1 minute of easy effort on a flat grade.

Because the OP asked the question so badly, that actually is the correct answer.

The question could make sense if it was asked in terms of distance instead of time.
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Old 07-25-22, 11:19 AM
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Originally Posted by KerryIrons View Post
Power to ride a bicycle can be calculated pretty accurately with a well known equation:horse power = [Vg*W(.0053 + %G/100) + .0083(Va^3)]/375
.
Excellent! Thank you!
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Old 07-25-22, 11:23 AM
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Originally Posted by KerryIrons View Post
Analyticcycling.com allows you to .
That site is dead.
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Old 07-25-22, 11:36 AM
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Originally Posted by livedarklions View Post
Because the OP asked the question so badly, that actually is the correct answer.

The question could make sense if it was asked in terms of distance instead of time.
You could get a powermeter and assign real numbers to this exercise if you wanted. Pedal your bike at 250 w (for example). You'd find out that 250 watts for 1 minute = 15 kj, whether you're going up a hill, or on the flat, or with a tailwind, or if you were on a heavy bike or a super-light time trial bike.
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Old 07-25-22, 02:01 PM
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Originally Posted by caloso View Post
You could get a powermeter and assign real numbers to this exercise if you wanted. Pedal your bike at 250 w (for example). You'd find out that 250 watts for 1 minute = 15 kj, whether you're going up a hill, or on the flat, or with a tailwind, or if you were on a heavy bike or a super-light time trial bike.
That's the problem with all these "X gives you the better workout" things, all increasing resistance does is increase the the effort per mile, and you can always just change the number of miles you ride either way to compensate.
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Old 07-25-22, 02:05 PM
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Originally Posted by caloso View Post
You could get a powermeter and assign real numbers to this exercise if you wanted. Pedal your bike at 250 w (for example). You'd find out that 250 watts for 1 minute = 15 kj, whether you're going up a hill, or on the flat, or with a tailwind, or if you were on a heavy bike or a super-light time trial bike.
The OP worded his question badly.
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Old 07-25-22, 02:08 PM
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1 mile flat = 180-200 vertical feet

YMMV depending on your parameters
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Old 07-25-22, 02:43 PM
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FYI, the last (aero) terms of each formula should be proportional to Vg * Va^2, rather than Va^3.

See, for example, https://www.sportsci.org/encyc/cyclin...wn.html#uphill

Otherwise, you would be needing power and doing work to stand still in a headwind.

Otto

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Old 07-27-22, 09:38 AM
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Originally Posted by ofajen View Post
FYI, the last (aero) terms of each formula should be proportional to Vg * Va^2, rather than Va^3.

See, for example, Cycling: Uphill and Downhill

Otherwise, you would be needing power and doing work to stand still in a headwind.

Otto
Yes. Thanks for that refinement. For most calculations people will assume ground speed and wind speed are the same, but obviously that will be wrong most of the time. That said, the actual wind speed that the rider experiences are hard to determine. My experience based on a large number of time trials is that the wind the rider sees is about 1/3 of what the weather is reporting. "Weather wind" is measured in the wide open and about 30 feet above the ground. Bushes, trees, buildings and simple ground effects all reduce the real speed at bike level.
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Old 07-27-22, 10:38 AM
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Originally Posted by KerryIrons View Post
That said, the actual wind speed that the rider experiences are hard to determine. My experience based on a large number of time trials is that the wind the rider sees is about 1/3 of what the weather is reporting. "Weather wind" is measured in the wide open and about 30 feet above the ground. Bushes, trees, buildings and simple ground effects all reduce the real speed at bike level.
Oh yeah. I mostly ride our trails and they are sheltered by trees in most places. In the warm part of the year, the leaves knock the wind down a good bit in many places.

In the winter, nearly all the leaves are gone and there is a pretty consistent 10-15 mph wind most every day and more in the early spring time before the leaves grow back.

Donít have the link handy but Jobst Brandt did a neat discussion on the effect of wind at all angles and basically itís a drag. A side wind increases the workload some, so averaged over all angles, the wind definitely slows you down or requires more power at speed.

Otto
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Old 07-27-22, 11:45 AM
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Originally Posted by GhostRider62 View Post
1 mile flat = 180-200 vertical feet

YMMV depending on your parameters

I see figures like that, but I have no idea what the basis is for that claim.
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Old 07-27-22, 12:10 PM
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Originally Posted by livedarklions View Post
I see figures like that, but I have no idea what the basis is for that claim.
Can be determined empirically or do the math.

550 foot pounds per second = the power of one good horse

An average cyclist is about as good as one third of a horse or about 245 watts.

A pretty fit 183 pound average recreational cyclist developing 1/3 of a horse could travel about 18-20 miles in an hour.....i do about 20 miles at that power on a normal road bike, nothing fancy. OTOH, if this same cyclist were to travel vertically, they would travel 3,606 feet for the same power (246 watts) required to travel 20 mph. So, we can say 20 miles is about 3600 feet or one mile is 180 vertical feet. Some riders are more aero. Some are heavier. But these are sort of my numbers at a tempo or high zone 2 pace when I am not fat. Hope this makes sense, I tried to use a lot of words. Lawyers like words.
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Old 07-27-22, 01:15 PM
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Originally Posted by GhostRider62 View Post
Can be determined empirically or do the math.


550 foot pounds per second = the power of one good horse


An average cyclist is about as good as one third of a horse or about 245 watts.


A pretty fit 183 pound average recreational cyclist developing 1/3 of a horse could travel about 18-20 miles in an hour.....i do about 20 miles at that power on a normal road bike, nothing fancy. OTOH, if this same cyclist were to travel vertically, they would travel 3,606 feet for the same power (246 watts) required to travel 20 mph. So, we can say 20 miles is about 3600 feet or one mile is 180 vertical feet. Some riders are more aero. Some are heavier. But these are sort of my numbers at a tempo or high zone 2 pace when I am not fat. Hope this makes sense, I tried to use a lot of words. Lawyers like words.

This is kind of interesting. I have pondered before how to do climbing / distance equivalencies and I struggle with it.


Looking at the last two rides I did... Saturday morning and last night. Both rides took me close to 90 minutes. I rode very similar effort and average heart rate on both. Although Ride 1 was done in pretty hot weather, and ride 2 was nice conditions.


Ride 1: 1890 feet of climbing, 101 feet of climbing per mile, 18.8 miles, 91 minutes, 12.4 mph average. Climbing via one mountain climb and one more decent sized hill. Route is 90% up or down. Very little flat, and no rollers either.

Ride 2: 1225 feet of climbing, 47 feet of climbing per mile, 26.3 miles, 86 minutes, 18.3 mph average. Climbing via many small hills, no big ones. Not much flat.


Looking at climbing to equivalent distance, the extra 665 feet I had to climb on Ride 1 reduced my distance by 7.5 miles. And if I had ridden for 5 more minutes on Ride 2 to equal Ride 1's time of 91 minutes, I would have gone another 1.5 miles. So, here the extra 665 feet penalized me by 9 miles. That's 74 feet of climbing = 1 mile.


While I can't fault your analysis, I am wondering what major factors might influence the equivalency. So that sometimes it's 180 ft = 1 mile on the flat, and sometimes its 74 ft climbing = 1 mile on the flat, and sometimes something else. Obviously aerodynamics and body/bike weight count, but perhaps it's also the nature of the climbing too that plays a role.
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Old 07-27-22, 01:17 PM
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Originally Posted by GhostRider62 View Post
Can be determined empirically or do the math.

550 foot pounds per second = the power of one good horse

An average cyclist is about as good as one third of a horse or about 245 watts.

A pretty fit 183 pound average recreational cyclist developing 1/3 of a horse could travel about 18-20 miles in an hour.....i do about 20 miles at that power on a normal road bike, nothing fancy. OTOH, if this same cyclist were to travel vertically, they would travel 3,606 feet for the same power (246 watts) required to travel 20 mph. So, we can say 20 miles is about 3600 feet or one mile is 180 vertical feet. Some riders are more aero. Some are heavier. But these are sort of my numbers at a tempo or high zone 2 pace when I am not fat. Hope this makes sense, I tried to use a lot of words. Lawyers like words.

Also a lawyer, so yeah.

Not a physicist here, but I'd have to think that would vary with grade. It's a lot easier to climb a foot at 1% grade than it is to climb the same foot at 25% grade. If the grade gets steep enough, it becomes actually impossible.
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Old 07-27-22, 01:53 PM
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'workouts' can be about more than 'work'

Originally Posted by caloso View Post
You could get a powermeter and assign real numbers to this exercise if you wanted. Pedal your bike at 250 w (for example). You'd find out that 250 watts for 1 minute = 15 kj, whether you're going up a hill, or on the flat, or with a tailwind, or if you were on a heavy bike or a super-light time trial bike.
True.

Originally Posted by livedarklions View Post
That's the problem with all these "X gives you the better workout" things, all increasing resistance does is increase the the effort per mile, and you can always just change the number of miles you ride either way to compensate.
False. Why do people believe that total energy expended is the only parameter relevant to describing a 'workout'?

For example, 10 repetitions with 200 lbs. is vastly different from 100 repetitions at 20 lbs., even if they both equal the same amount of work. Why do so many insist otherwise?
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Old 07-27-22, 01:58 PM
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Originally Posted by Fredo76 View Post
True.

False. Why do people believe that total energy expended is the only parameter relevant to describing a 'workout'?

For example, 10 repetitions with 200 lbs. is vastly different from 100 repetitions at 20 lbs., even if they both equal the same amount of work. Why do so many insist otherwise?
Because most people don't understand the concept of training stress.
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Old 07-27-22, 02:10 PM
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Originally Posted by Steamer View Post
This is kind of interesting. I have pondered before how to do climbing / distance equivalencies and I struggle with it.


Looking at the last two rides I did... Saturday morning and last night. Both rides took me close to 90 minutes. I rode very similar effort and average heart rate on both. Although Ride 1 was done in pretty hot weather, and ride 2 was nice conditions.


Ride 1: 1890 feet of climbing, 101 feet of climbing per mile, 18.8 miles, 91 minutes, 12.4 mph average. Climbing via one mountain climb and one more decent sized hill. Route is 90% up or down. Very little flat, and no rollers either.

Ride 2: 1225 feet of climbing, 47 feet of climbing per mile, 26.3 miles, 86 minutes, 18.3 mph average. Climbing via many small hills, no big ones. Not much flat.


Looking at climbing to equivalent distance, the extra 665 feet I had to climb on Ride 1 reduced my distance by 7.5 miles. And if I had ridden for 5 more minutes on Ride 2 to equal Ride 1's time of 91 minutes, I would have gone another 1.5 miles. So, here the extra 665 feet penalized me by 9 miles. That's 74 feet of climbing = 1 mile.


While I can't fault your analysis, I am wondering what major factors might influence the equivalency. So that sometimes it's 180 ft = 1 mile on the flat, and sometimes its 74 ft climbing = 1 mile on the flat, and sometimes something else. Obviously aerodynamics and body/bike weight count, but perhaps it's also the nature of the climbing too that plays a role.
Admit that this is a rough way to look at a ride.

So, we're both randonneurs. Let's say I have a 200K with 7,500 feet of climbing. The distance takes me around 6:15. The elevation takes me about 2:15. So, it is going to be a little under 9 hours with controls. If it is a flatter course like in southern NJ or Florida. It is going to be under 7 hours.

On a 300K recently, there was about 11,500 feet of climbing. I rode hard and did under 13 hours. About 20 mph for the distance or about 9 hours and a little under 4 hours for the climbing.

Where it gets tricky is on a long ride like a 1200k because of fatigue and 100 feet might be more like a mile after the first day or so. I predicted my PBP time within 30 minutes with this approach with slight modification, I ignore short shallow hills and because they will be steam rolled.

Note: This back of the envelope is for my upright. The bent is more like 25 mph on the flats and less elevation per hour. I wear the same kit on my upright, so, the aerodynamics are about the same on any brevet. The bent aerodynamics are in another world so to speak. Again, just a rough way for an approximate equivalency, It can be important if there is a control at the top of a mountain......I just barely made the second control out West due to a big climb and several riders just missed it. I knew based on elevation that it would be tight and rode hard, just barely making it.

Another factor is whether you ride at the same output from ride to ride. I tend to ride at a certain HR and it corresponds to a certain power (zone 2). If I am fit and healthy, this power starts at a certain level and declines as time goes by and unless it is very hot, this decline is also fairly predictable.

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Old 07-27-22, 02:12 PM
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Originally Posted by Fredo76 View Post
True.

False. Why do people believe that total energy expended is the only parameter relevant to describing a 'workout'?

For example, 10 repetitions with 200 lbs. is vastly different from 100 repetitions at 20 lbs., even if they both equal the same amount of work. Why do so many insist otherwise?
Because bicycles aren't weights being lifted so the analogy is completely wrong--the differences in resistance between the lighter and heavier bikes are not so significant as to affect the effects of resistance meaningfully. This is more like comparing lifting 198 pounds 11 times to lifting 200 pounds 10 times, but that's not even true as you'll see below.

You aren't the first person to misapply weightlifting logic here, Start with the fact that the differences in weight between bikes are likely small in comparison to the gross weight of the rider and the bike, and that the bike is actually a machine designed to minimize the vertical lifting of the weight and to maximize the advantages of forward momentum and you'll soon realize that the analogies are more confusing than they are helpful.

Also, for that matter, riding in a higher gear, or riding faster add resistance (both mechanical and wind).. Increased weight also just adds resistance. And as pointed out repeatedly, if you really want your resistance in the form of weight, just carry something heavy on your light bike.
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Old 07-27-22, 02:18 PM
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Originally Posted by caloso View Post
Because most people don't understand the concept of training stress.

Wrong. There's no reason you can't get as much training stress from a lighter bike, one simply adjusts gears, routes, payload, and speed. And if you really want to do it with weight, carry something on your bike.

Want to do the math? How much faster do you have to ride a bike that's 5 pounds lighter in order to equalize the resistance that's being overcome? Please be sure to include drag in your calculations.

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