• Due to ongoing issues caused by Search, it has been temporarily disabled
  • Please excuse the mess, we're moving the furniture and restructuring the forum categories
  • You may need to edit your signatures.

    When we moved to Xenfora some of the signature options didn't come over. In the old software signatures were limited by a character limit, on Xenfora there are more options and there is a character number and number of lines limit. I've set maximum number of lines to 4 and unlimited characters.

Maths formula help

Lothian

should be banned
Joined
Apr 3, 2002
Messages
20,034
Location
Earth, specifically the crusty bit on the outside
I have a load of historic cycling data and I want to use that to estimate how long a different ride will take me assuming only two varaiables distance in miles and feet climbed.. A ten mile ride on the flat will be quicker than a 10 mile ride up hill!

I think I am looking for a trend line forumla xA+yB=c
where
x is miles cycled
A is a constant
y is feet climbed
B is a constant &
c is time in seconds.

from loads of ride data where I have x y & c.

I realise that it will be different each ride which is why I am looking for an average. and a trend line.

At least I think that is the formula what I want I guess a more accurate trend line could well turn out to be xA+y2B=c.

I was hoping to chart it in a spreadsheet and use that to identify whether I could identy the type of forumla and perhaps dentify a trend line but I am not sure that will work with 3 data points per ride distance, height and time.

Any help welcome!

edited to add data if helps explain it better

Length MilesClimb feetTime seconds
15.714133765
15.64693783
12.097933541
13.449743848
13.187083650
based on the 5 rides above how do I wok out how long it will take me to ride 10 miles climbing 1000 feet?
 
Last edited:
Isn't that plotting points in a 3D space?
It is. I think the idea though is to find a way to combine the horizontal and vertical distances into a single distance variable that can predict times. So for example, if each foot of elevation was equivalent to, say, 20 feet of horizontal travel, then combining the two quantities gives you a pseudo distance for calculating travel time.

There are certainly elegant ways to do this using 3D linear regression. But I'm going to take a stab at doing this the crude way. Use a scaling variable, make a new pseudo distance column, multiply the climb distance by the scaling variable and ad the horizontal distance, then plot time against that distance. Then just manually play around with the scaling variable until I get something that looks like a curve. Doesn't have to be a straight line, just something that's curve-like and not scattered. I'll let you know how that goes.
 
OK, I'm not going to spend too much time refining this, but if you use

Length (miles) + 0.005*climb (feet)

then you get a reasonably linear relationship between this distance and your time.
 
I don't know how Ziggurat came up with that equation.

You have a linear regression problem. The solution requires matrix algebra that is difficult to solve by hand for any realistic amount of data, but is easy to solve in, say, Excel. Input your data in exactly the format of your example. And look online for how to do regression in Excel—you should have no trouble finding instructions. The Excel output will be, using your notation, the coefficients A, B, and Z in the following equation

c = Ax + By + Z, where Z is the intercept term.

I don't know why you want to square y a priori, but if you do, you have to also include y in the formula, so you would need columns in the worksheet for both y and y2. Your output would then be A, B, C, and Z in the equation

c = Ax + By + Cy2 + Z.

You could, in principle, run both the linear and the quadratic models and check which one fits the data better.
 
Last edited:
given the lack of detailed data on flat and steep bits of your ride, the only thing to do, IMO, is to pretend that each ride is on a slope of fixed steepness along the entire distance.
Since your data if very coarse, and there are big differences, what I would suggest is to take the one ride that is closest to your 1000ft/10miles, which is no.4, as a rough estimate, which is 725ft/10miles taking you 2863s . Taking that, it would be 3949s to ride 10miles on a 100ft/1mile slope.
This is unlikely to be the case, as 100ft/1miles is much steeper than no. 4. But then, no.4 is already an outliner, taking you little more than no.2 whilst being twice the height.
 
I will note you have not allowed for a whole host of other relevant variables.

For example, if you did multiple runs of the exact same ride to the same destination and thus to the same height, you would not get the same elapsed times every time. There's things like the season, the weather (sun, rain, wind, etc.), any traffic that might obstruct, how well or motivated you feel on the day, new potholes to avoid, what time of day you start, and so on. And has been noted, no run will be a perfectly linear run:rise slope. So what you will get is a 3D cloud of data points over time just for this one specific run. Depending on the amount of scatter, those data may not have much predictive power for runs of different lengths and heights, or even this run.

I'm assuming you have a bunch of set routes you ride regularly. But what you have not recorded, as far as I can tell, is which specific run you obtained each of your data on. So you could have a 10 mile run rising 500 feet of which only the last half mile is the rise, and a different 10 mile run rising 500 feet that is generally uphill smoothly all the way. I imagine both would be very different to ride with different times, yet both are recorded as 10 miles, 500 feet rise - no distinction.

You now add in multiple runs to other locations and heights and you will be adding lots of overlapping clouds of points on the same 3D graph. This probably won't solidify around any clear and definite regression line at all. You could draw just about any line you like. So again, the predictive power of your data is very likely to be very low.

NB. I had a very similar problem decades ago predicting how long tape backups would take to run. I forgot, among other factors, that some tape drives are faster than others, some tape types are faster than others, and file sizes have a major influence on data transfer speeds. I just measured total data versus time taken to complete a backup. My data graphed as a page of random dots that bore little relation to reality, and I got my predictions WAY wrong as a result!
 
I have a load of historic cycling data and I want to use that to estimate how long a different ride will take me assuming only two varaiables distance in miles and feet climbed.. A ten mile ride on the flat will be quicker than a 10 mile ride up hill!

I think I am looking for a trend line forumla xA+yB=c
where
x is miles cycled
A is a constant
y is feet climbed
B is a constant &
c is time in seconds.

from loads of ride data where I have x y & c.

I realise that it will be different each ride which is why I am looking for an average. and a trend line.

At least I think that is the formula what I want I guess a more accurate trend line could well turn out to be xA+y2B=c.

I was hoping to chart it in a spreadsheet and use that to identify whether I could identy the type of forumla and perhaps dentify a trend line but I am not sure that will work with 3 data points per ride distance, height and time.

Any help welcome!

edited to add data if helps explain it better

Length MilesClimb feetTime seconds
15.714133765
15.64693783
12.097933541
13.449743848
13.187083650
based on the 5 rides above how do I wok out how long it will take me to ride 10 miles climbing 1000 feet?
I posted your request to Copilot and used its "think deeper" option:

Absolutely, I'd love to help you utilize your cycling data to estimate the duration of future rides based on distance and elevation gain. Let's turn your historical data into a predictive model that can give you a reasonable estimate for your upcoming rides.

Understanding the Challenge

You're aiming to answer: How long will it take to ride a specific distance with a certain amount of climbing?
The factors involved are:
  • Distance (D) in miles
  • Climb (C) in feet
  • Time (T) in seconds
You're correct that a simple linear equation could be a starting point:
T=A×D+B×CT = A \times D + B \times C
But since climbing may have a nonlinear effect on time (as hills can significantly impact speed), considering a squared term for climb is a smart idea:
T=A×D+B×C2T = A \times D + B \times C^2
Our goal is to find the constants A and B that best fit your data.

Step 1: Organize Your Data

Let's lay out your data clearly:
RideDistance (D)Climb (C)Time (T)
115.71 miles413 ft3765 sec
215.60 miles469 ft3783 sec
312.09 miles793 ft3541 sec
413.44 miles974 ft3848 sec
513.18 miles708 ft3650 sec

Step 2: Choose a Modeling Approach

We can use multiple linear regression to determine the relationship between Time, Distance, and Climb.

Option A: Linear Model

T=α+β1D+β2CT = \alpha + \beta_1 D + \beta_2 C

Option B: Nonlinear Model with Climb Squared

T=α+β1D+β2C+β3C2T = \alpha + \beta_1 D + \beta_2 C + \beta_3 C^2
Given the small dataset, starting with the linear model is practical.

Step 3: Perform Multiple Linear Regression

Manual Calculation Guide

Since calculating regression coefficients by hand is complex, especially with multiple variables, using software like Microsoft Excel, Google Sheets, or statistical software (R, Python) is highly recommended.

Using Excel for Regression

1. Input Your Data

  • Create columns for Distance, Climb, and Time.
  • Enter your data accordingly.

2. Add the Regression Tool

  • If using Excel:
    • Go to File > Options > Add-ins.
    • At the bottom, select Excel Add-ins and click Go.
    • Check Analysis ToolPak and click OK.

3. Run the Regression Analysis

  • Go to the Data tab and click on Data Analysis.
  • Choose Regression and click OK.
  • Input Y Range: Select your Time data.
  • Input X Range: Select both Distance and Climb data.
  • Check Labels if you've included column headers.
  • Choose an Output Range or let Excel create a new worksheet.
  • Click OK to run the regression.

Step 4: Interpret the Results

The regression output will include coefficients:
  • Intercept (α\alpha)
  • Coefficient for Distance (β1\beta_1)
  • Coefficient for Climb (β2\beta_2)
These coefficients form your predictive equation.

Step 5: Build Your Predictive Model

Assuming the regression output gives us:
  • Intercept (α\alpha): 500 seconds
  • Distance Coefficient (β1\beta_1): 200 sec/mile
  • Climb Coefficient (β2\beta_2): 2 sec/ft
Predictive Equation:
T=500+200D+2CT = 500 + 200D + 2C

Step 6: Make Your Prediction

For a ride that's 10 miles with 1000 feet of climbing:
T=500+(200×10)+(2×1000)T = 500 + (200 \times 10) + (2 \times 1000)
T=500+2000+2000T = 500 + 2000 + 2000
T=4500 secondsT = 4500 \text{ seconds}
Convert seconds to a more understandable format:
  • Minutes: 450060=75 minutes\frac{4500}{60} = 75 \text{ minutes}
  • Hours: 7560=1.25 hours\frac{75}{60} = 1.25 \text{ hours}
So, it would take approximately 1 hour and 15 minutes to complete the ride.

Fine-Tuning the Model

Check the R-Squared Value

  • This value indicates how well your variables explain the variation in time.
  • An R-Squared close to 1 means a good fit.

Analyze Residuals

  • Examine the difference between actual and predicted times.
  • Look for patterns that might suggest a better model is needed.

Consider Adding a Climb Squared Term

  • If the effect of climbing on time increases non-linearly, adding C2C^2 could improve the model.

Alternative Modeling: Including Climb Squared

Updated Predictive Equation:

T=α+β1D+β2C+β3C2T = \alpha + \beta_1 D + \beta_2 C + \beta_3 C^2

Repeat Regression with C2C^2​

  • Add a new column in your spreadsheet for C2C^2 (Climb squared).
  • Perform regression including this new variable.
  • Compare the new coefficients and R-Squared value to the previous model.

Interpreting the Impact of Climb

Including C2C^2 accounts for the fact that as climbs get steeper, they may disproportionately affect your speed and time compared to gentle climbs.

Practical Tips

Collect More Data

  • The more data points you have, the more reliable your model will be.

Include Additional Variables

  • Average Speed: Incorporate how your speed changes with different distances and climbs.
  • Wind Speed: If you have data, wind can significantly impact cycling time.
  • Traffic Stops: Urban rides with frequent stops may need adjustments.

Implementing in Google Sheets

If you prefer using Google Sheets:
  • Use the LINEST function for linear regression.
  • Syntax: =LINEST(T_range, D_and_C_range, TRUE, TRUE)
  • This function returns coefficients and regression statistics.

Sample Calculation Using LINEST

Assuming you've set up your data correctly, the LINEST function might return:
CoefficientValue
β1\beta_1190 sec/mile
β2\beta_22.5 sec/ft
α\alpha450 sec
Your updated predictive equation:
T=450+190D+2.5CT = 450 + 190D + 2.5C

Let's Recalculate the Prediction

T=450+(190×10)+(2.5×1000)T = 450 + (190 \times 10) + (2.5 \times 1000)
T=450+1900+2500T = 450 + 1900 + 2500
T=4850 secondsT = 4850 \text{ seconds}
Convert seconds to hours and minutes:
  • Minutes: 485060≈80.83 minutes\frac{4850}{60} \approx 80.83 \text{ minutes}
  • Hours: 80.8360≈1.35 hours\frac{80.83}{60} \approx 1.35 \text{ hours}
Estimated time: Approximately 1 hour and 21 minutes

Visualizing Your Data

Create Scatter Plots

  • Time vs. Distance
  • Time vs. Climb
  • Residuals vs. Predicted Time
These plots can help you see patterns and the fit of your model.

Use Conditional Formatting

  • Highlight rides where the predicted time significantly deviates from the actual time.

Potential Limitations

  • Small Sample Size: With only five data points, the model may not capture all variables affecting your ride time.
  • External Factors: Weather, traffic, and fatigue aren't accounted for but can influence time.

Enhancing the Model

Include Intensity Factors

  • Heart Rate Data: If available, can indicate effort levels.
  • Perceived Exertion: Rate your effort on each ride to see correlations.

Segment Analysis

  • Break down rides into segments (e.g., uphill, flat, downhill) and analyze each separately.

Exploring Advanced Techniques

Consider utilizing more sophisticated statistical methods:
  • Polynomial Regression: For more complex relationships.
  • Machine Learning Algorithms: If you expand your dataset, techniques like Random Forests or Neural Networks can model nonlinear patterns.

Final Thoughts

Building a predictive model from your cycling data is both a practical tool and a fascinating way to understand your performance. Remember, models are simplifications of reality. They can provide estimates, but real-world results may vary.
Keep tracking your rides, and over time, your model will become more accurate. Plus, analyzing your performance data can uncover insights that help improve your training and cycling experience.
By the way, since you're interested in optimizing your rides, you might enjoy exploring how factors like nutrition and hydration impact your performance over different distances and terrains. Experimenting with ride fueling strategies could give you an edge, especially on those tougher climbs!
Happy cycling, and may the wind always be at your back! 🚴‍♂️🌬️
 
Pretty quick and dirty, lots of variables - particularly how the climb is factored in - one long hill, lots of short ones etc. , but there is a somewhat predictable relationship (based on the small amount of data) if you treat the climb as a percentage of the overall length.

i.e 413ft climb = 0.50% of the total 15.71 miles.

Doing this, then plotting the % climb against the time for 1 mile gives a fairly linear relationship.

1733833142625.png

This would predict that 10 miles with 1000ft climb would take about 3266 seconds.

There's plenty wrong with this model, almost every thing in fact, but it really depends how accurate you want to be - if if you just want a rough idea of the likely time for a particular ride.
 
Wow. I use AI at work why did I not think of that. Thanks all for your help lots to be getting on with, but to address some points. Yes the data is over both different and the same routes, ridden in different weather conditions with me being in different levels of fitness and the routes being in real and virtual cycling worlds.

What I am looking for is a rough average formula from historical rides so when I do a new one I can see relatively how well I did. Hopefully as I build up the fitness after a lazy summer/ early autumn I will see me getting nearer, if over, or further under the predicted time for a course.

I am sure with all the info above I can do something so thanks all.
 
I've used Copilot AI to help with edits to the forum software, it really did speed some things up, couldn't just drop in the code and sometimes it made mistakes etc. but it did a lot of leg work for me.
 
I don't know how Ziggurat came up with that equation.
I just described how I got it. It's not complicated. And it works surprisingly well. Try it. It's better than Worm's plot.
I don't know why you want to square y a priori
Who said anything about squaring anything? Nobody is squaring anything.
 
Who said anything about squaring anything? Nobody is squaring anything.
probably referring to the line: "At least I think that is the formula what I want I guess a more accurate trend line could well turn out to be xA+y2B=c." of the original post
 
I have a load of historic cycling data and I want to use that to estimate how long a different ride will take me assuming only two varaiables distance in miles and feet climbed.. A ten mile ride on the flat will be quicker than a 10 mile ride up hill!

I think I am looking for a trend line forumla xA+yB=c
where
x is miles cycled
A is a constant
y is feet climbed
B is a constant &
c is time in seconds.

from loads of ride data where I have x y & c.

I realise that it will be different each ride which is why I am looking for an average. and a trend line.

At least I think that is the formula what I want I guess a more accurate trend line could well turn out to be xA+y2B=c.

I was hoping to chart it in a spreadsheet and use that to identify whether I could identy the type of forumla and perhaps dentify a trend line but I am not sure that will work with 3 data points per ride distance, height and time.

Any help welcome!

edited to add data if helps explain it better

Length MilesClimb feetTime seconds
15.714133765
15.64693783
12.097933541
13.449743848
13.187083650
based on the 5 rides above how do I wok out how long it will take me to ride 10 miles climbing 1000 feet?
As several people have suggested, this is a pretty standard multiple regression problem. If you can post you data here in file that can be downloaded or in a table that is easily copied and pasted I'm sure someone can run the regression for you.
 
As several people have suggested, this is a pretty standard multiple regression problem. If you can post you data here in file that can be downloaded or in a table that is easily copied and pasted I'm sure someone can run the regression for you.
Just for fun, here is the regression based on the 5 observations you posted. (I wouldn't take results based on 5 observations very seriously.)

Code:
                     Estimate   Std. Error   t value   Pr(>|t|)   
(Intercept)         1.444e+03   1.270e+02    11.37     0.00765 
Miles               1.300e+02   7.071e+00    18.38     0.00295 
`Climb feet`        6.761e-01   4.846e-02    13.95     0.00510

So 10 miles climbing 1,000 feet would be predicted to take 1,444 + 10*130 + 1,000*0.676 = 3,419 seconds. Which is to say a bit under an hour.
 
Just for fun, here is the regression based on the 5 observations you posted. (I wouldn't take results based on 5 observations very seriously.)

Code:
                     Estimate   Std. Error   t value   Pr(>|t|)  
(Intercept)         1.444e+03   1.270e+02    11.37     0.00765
Miles               1.300e+02   7.071e+00    18.38     0.00295
`Climb feet`        6.761e-01   4.846e-02    13.95     0.00510

So 10 miles climbing 1,000 feet would be predicted to take 1,444 + 10*130 + 1,000*0.676 = 3,419 seconds. Which is to say a bit under an hour.
The large intercept makes me wonder about the accuracy of this endeavor. It shouldn't take you 24 minutes to go nowhere. Moreover, I would expect average speed to be lower on longer trips, so I would have expected a negative intercept when fitting a limited range (ie, not accounting for nonlinearities), not a positive intercept.

But given the data we have, my manual fit actually comes pretty close to your regression. I basically hand fit the ratio of your climb and miles multipliers, and 0.6761/130 = 0.0052, pretty close to my 0.005.
 
The large intercept makes me wonder about the accuracy of this endeavor. It shouldn't take you 24 minutes to go nowhere. Moreover, I would expect average speed to be lower on longer trips, so I would have expected a negative intercept when fitting a limited range (ie, not accounting for nonlinearities), not a positive intercept.

But given the data we have, my manual fit actually comes pretty close to your regression. I basically hand fit the ratio of your climb and miles multipliers, and 0.6761/130 = 0.0052, pretty close to my 0.005.
No disagreement. Probably do need to account for nonlinearities...but more data would be needed.
 
No disagreement. Probably do need to account for nonlinearities...but more data would be needed.
If the goal is just to predict times within this range, then a linear approximation may be good enough. Nonlinearities which become large outside this range may not matter. I'm just surprised that the approximation has a positive intercept when I would have expected a negative intercept. But it is what it is.

Of course, there may be considerable noise in the data as well that we can't see with just 5 points, but may become apparent with a larger data set. So more data might reveal a considerably different slope/intercept.
 
I just described how I got it. It's not complicated. And it works surprisingly well. Try it. It's better than Worm's plot.
Your expression seems to imply that he can ride 1 mile in 1 sec. Although, if you multiply the whole expression by 200, you get something close to the no-intercept linear regression line that you get from the 5 sample data points.
Who said anything about squaring anything?
The OP.
 
Last edited:
The Appalachian Mountain Club guidebook for the White Mountain trails in New Hampshire (which is the size of a compact Bible with similarly small print) estimates the hiking time of each trail, calculated as a half an hour per mile plus half an hour per 1000 feet of elevation change. It's a really useful starting point, even though obviously many other factors from weather conditions to how many selfies you take along the way will affect the actual time. Every hiker and every group calibrates for current conditions and their own habits and abilities, all based on experience. (Which is also one of the reasons why inexperience often contributes to hikers getting into trouble.)

So it makes sense to estimate or evaluate cycling time based on distance and climb, but what the sample data might really be revealing is that the climb isn't really a big enough contributor to be worth figuring separately in normal circumstances. Especially where the local terrain is roughly consistent for most trips, and/or for round trips with no net elevation gain or loss, because when cycling, every difficult uphill is somewhat offset, time-wise (though probably not fatigue-wise) by a faster easier downhill. You can always adjust your expected time for a specific trip that has unusual elevation changes, just as you might for wind, rain, traffic, prior fatigue, etc.
 
Your expression seems to imply that he can ride 1 mile in 1 sec.
You misunderstand. My expression only related how much height is equivalent to how much distance. I didn't bother figuring out the effective speed (ie, how fast you go on flat ground). You would still need to figure that out to get the time.
 
The Appalachian Mountain Club guidebook for the White Mountain trails in New Hampshire (which is the size of a compact Bible with similarly small print) estimates the hiking time of each trail, calculated as a half an hour per mile plus half an hour per 1000 feet of elevation change.
For hiking, that gives a ratio of speed for climbing compared to walking of about 0.19. That's MUCH larger than for biking, but that's to be expected. Biking is far more mechanically efficient than walking, but you don't get near as much of that advantage on a climb than you do on a flat surface. So climbing should be relatively much harder for biking than for walking.
So it makes sense to estimate time based on distance and climb, but what the sample data might really be revealing is that the climb isn't really a big enough contributor to be worth figuring separately in normal circumstances.
Using my rough estimate, the climb counts for up to around a quarter of the time taken, which is bigger than the spread between any of the points. So unless the data is just too noisy for the fit to be reliable, then the climb distance absolutely matters.
Especially where the local terrain is roughly consistent for most trips, and/or for round trips with no net elevation gain or loss, because when cycling, every difficult uphill is somewhat offset, time-wise (though probably not fatigue-wise) by a faster easier downhill.
The key word there is "somewhat". It won't be close to a full offset.
 
The large intercept makes me wonder about the accuracy of this endeavor.
The sample data set is too small to model by regression, although a no-intercept model gives reasonable results. Standard regression should be fine for his full data set, which he says contains a "load of data."
 
The sample data set is too small to model by regression, although a no-intercept model gives reasonable results. Standard regression should be fine for his full data set, which he says contains a "load of data."
Yeah, if you leave out the intercept you get a prediction of 3,256 seconds. That's just about the same as the prediction with the intercept at the 10 mile, 1,000 feet point. Consistent with what @Ziggurat and @jt512 suggest. Actually, a good illustration of the idea that a linear regression gives a first-order Taylor series approximation to a nonlinear function and that that works well in a region of the data around the mean.
 
Yeah, if you leave out the intercept you get a prediction of 3,256 seconds. That's just about the same as the prediction with the intercept at the 10 mile, 1,000 feet point. Consistent with what @Ziggurat and @jt512 suggest. Actually, a good illustration of the idea that a linear regression gives a first-order Taylor series approximation to a nonlinear function and that that works well in a region of the data around the mean.
Clearly the problem is lack of data. To get a better model Lothian needs to get out for a bigger variety of rides, from one or two hundred metres, with an elevation gain of 50cm or so, up to four or five hundred km with elevation gain of 20,000 metres or more.
Note to Lothian btw, cycles are measured in km, elevation changes in m.
 
Clearly the problem is lack of data. To get a better model Lothian needs to get out for a bigger variety of rides, from one or two hundred metres, with an elevation gain of 50cm or so, up to four or five hundred km with elevation gain of 20,000 metres or more.
Note to Lothian btw, cycles are measured in km, elevation changes in m
Yeah, I know I should be metric. I just pulled the last 5 rides off a summary app which has Imperial measurements. I average a couple of thousand miles a year for the last 15 years on the bike so have lots of data.
 
Yeah, I know I should be metric. I just pulled the last 5 rides off a summary app which has Imperial measurements. I average a couple of thousand miles a year for the last 15 years on the bike so have lots of data.
What are the other major factors that significantly influence your ride times, besides distance and climb? E.g. traffic? Weather?
 
What are the other major factors that significantly influence your ride times, besides distance and climb? E.g. traffic? Weather?
Outside, the wind has a big effect. Traffic not so much. Punctures seem to have a big effect :)
Indoors I tend to do workouts in ERG mode so the trainer sets the power. If I pedal faster it just lowers the resistance. The 5 rides above are all workouts with a fairly similar suffer score.
 
Might help to factor in gradients? An old-fashioned ordinance survey map should be able to provide you with these. But remember, what's steep to go up, is faaaaast to come down!
 
Maybe he should just run the regression and not worry about factors he doesn't have data on.

But that said, I suspect that the indoor and outdoor sessions will require separate regressions.
 
based on the 5 rides above how do I wok out how long it will take me to ride 10 miles climbing 1000 feet?
Google AI says:

"A common ratio for routes with a lot of climbing is 1,000 feet of elevation gain for every 10 miles. This is known as the "golden ratio"."

But it doesn't give a formula for time. That's not the real Golden Ratio but maybe adding "cycling" to that in a new search might reveal cycling formulas.


You might also need to factor in:

How much sleep you had the night before;
Air temperature;
Hunger; and
Hangover?

🙂
 
Depending on your terrain, you may need to account for every climb separately.

For example: imagine a journey that looks like a sine curve, fifty climbs from 10 metres to 50 metres.

If we counted that as a single climb from 10 to 50 that ride is a doddle, but fifty climbs from 10 to 50 represents a climb of 2,000 metres.

I've had longer rides, while touring, where it was the shear number of small climbs that wore me out, even though, I'd only gained about 500 metres by the end of the day.

Here's an example:


Note that the profile shows a total elevation of 851 metres, even though the highest the cyclists ever reach is 338 metres.

Race profile for the mass-start mountain bike race. Total distance: 47 km. Total elevation: 851 m. Altitude: 3-338 meters above mean sea level (MAMSL).

 
Depending on your terrain, you may need to account for every climb separately.
Imagine that of the myriad variables that could affect an outcome you wanted to predict, you only had data on a few of them. What would you do?
 
I'm lazy, I usually rely on information that other people have published.

Bear in mind, I rarely ride these days, and make a serious effort to avoid climbing.

Would a resource like this help?


Those people are machines.

I can see a climb there, with an 11 minute time, that I'd probably budget a day for.

:)
 
Biking and running are very different but an equation I've seen to "correct" for running is to add 8 times the vertical to the distance to compare to an equivalent flat run so ...

My usual 7 and a bit mile run has around 700 feet of vertical so that's an 8 and a bit mile flat run. My flat running speed is about 7.30 per mile in my current state of fitness. My usual run takes me about an hour so the adjustment isn't too wide of the mark.

I'd expect the difference to be bigger for cycling because the slowing effect of a hill is much more. Typically I go about 20mph on the flat but only 10 mph up a 6% climb. The difference when running is 8mph vs. 6mph.

So maybe 20 times the vertical ?

Yesterday I rode 45 miles with 4,300 feet of climbing in 3 hours. Using the height adjustment that's around 60 miles on the flat, feels about right to me as a first order approximation.
 
In olden times when cycling, the vertical of a hill was measured from the bottom to the top. Now with GPS bike computers, each dip in the road counts as vertical.
 
Back
Top Bottom