Friday, December 19, 2014

W/m^2, Altitude and the Hour Record. Part II

The Physics recap

In an earlier blog post I examined the influence of altitude on the physics of cycling’s world hour record, and showed how the reduction in air density as altitude increases means one can travel faster for the same power output, or put another way, the power demand reduces at any given speed as altitude increases.
That resulted in this chart, which shows the relationship in power to drag ratio (W/m^2) for speeds ranging from 47km/h up to Chris Boardman's 56.375km/h record. I've slightly amended the chart to extend up to altitude of 3,000 metres. Click on the pic to see a larger version.

Each slightly curved coloured line represents a speed as marked, and from that you can see how the W/m^2 required reduces with increasing attitude. The chart clearly suggests there is an advantage to performing such record attempts at higher altitudes, but it's never that simple of course. 

And the Physiological impact...

As we climb to higher altitudes and air density drops, the "thinner" air also means a reduction in the partial pressure of oxygen, which negatively impacts the power output we can sustain via aerobic metabolism. That loss of power can be as much as 20% or more depending on how high we go, and our individual response to altitude.

So the gain in speed from the physics side of the equation is somewhat negated by the reduction in physiological capacity. But by how much, and what might be the optimal or "sweet spot" altitude for a cyclist seeking to set a new record?

The physics side of the equation is easier to predict than the physiological, since the physics applies equally to all, however individual physiological response to altitude is quite variable, both from person to person, but also depending on how well a rider has acclimated to altitude. There are even differences in how altitude affects elite versus non-elite riders.

There have been a few published papers examining the impact of altitude on aerobic athletic performance and from these formulas to estimate the loss of power as a function of altitude have been developed. There was one from the 1989 paper by Peronnet et al, two from the 1999 paper by Bassett et al, one each for acclimated and non-acclimated athletes. Adding to those, I have generated a fourth formula, based on the 2007 study by Clark et al. The relevant papers are:

Péronnet F, Bouissou P, Perrault H, Ricci J.:
A comparison of cyclists' time records according to altitude and materials used.

Bassett DR Jr, Kyle CR, Passfield L, Broker JP, Burke ER.:
Comparing cycling world hour records, 1967-1996: modelling with empirical data.
Clark SA, Bourdon PC, Schmidt W, Singh B, Cable G, Onus KJ, Woolford SM, Stanef T, Gore CJ, Aughey RJ.:
The effect of acute simulated moderate altitude on power, performance and pacing strategies in well-trained cyclists.

Peronnet et al used empirical data from actual world cycling hour records to estimate the impact of altitude on an elite cyclist's power output. The assumptions used in estimating altitude induced power loss may have some error; in particular due to methods used to estimate the power for each rider as neither the power nor coefficient of aerodynamic drag was actually measured.

According to the old Wattage forum FAQ item by Dr David Bassett, Jr, the two Bassett et al formula were derived from earlier papers examining altitude impact on aerobic performance of four groups of highly trained or elite runners. So while these formulas were not derived from cyclists we can still generalise from those to the loss of aerobic capacity for cyclists.

Finally, the study by Clark et al measured the impact on peak oxygen utilisation (VO2), gross efficiency and cycling power output on ten well trained but non-altitude acclimated cyclists and triathletes by testing riders at simulated altitudes of 200, 1200, 2200 and 3200 metres. They examined a number of factors, including maximal 5-minute power output, VO2 and gross efficiency relative to performance at 200 metres, as well as sub-maximal VO2 and gross efficiency.

I used these data to generate a formula similar to those from Peronnet et al and Bassett et al. Of course there is an assumption of an equivalent reduction in 1-hour power as for 5-minute power. Clark et al noted slightly greater reductions in VO2 peak than for 5-minute maximal power, and no change in gross efficiency at 5-min max power with altitude. So there is some anaerobic metabolic contribution presumably making up the difference. There was some loss of sub-maximal efficiency noted at a simulated 3200 metres.

I chose in this instance to use the reduction in 5-minute power rather than fall in VO2 peak as the base data for the formula, and applied an adjustment to offset the formula for sea-level equivalency to bring it into line with the formula by Peronnet et al and Bassett et al. Of course when you look at the reported data there are of course sizeable variations within the test group at each simulated altitude, so the formula is based on group averages for each simulated altitude.

Here are the formulas:

x = kilometres above sea level:
Peronnet et al:           
Proportion of sea level power = -0.003x3 + 0.0081x2 - 0.0381x + 1
Bassett et al Altitude-acclimatised athletes (several weeks at altitude):
Proportion of sea level power = -0.0112 x2 – 0.0190x + 1
R2 = 0.973
Bassett et al Non altitude-acclimatised athletes (1-7 days at altitude):
Proportion of sea level power  = 0.00178x3 – 0.0143x2 – 0.0407x + 1
R2 = 0.974
Simmons’ formula based on Clark et al:
Proportion of sea level power  = -0.0092x2 – 0.0323x + 1
R2 = 0.993

So how do each of these estimates of power reduction at altitude compare? Well here's a plot of these formula:

There is some variance between each formula's estimates, although the gap between the Non-acclimatised athlete estimates by Bassett et al and by Simmons based on Clarke et al is not all that large, ranging up to a ~2% variance. 

Had I chosen to use the reduction in peak VO2 for 5-min max power, then I'd expect those two lines to be closer. In any case, these data by Clark et al reasonably match earlier reported findings of the impact of altitude on sustainable aerobic power. And once again - the individual response varies - these are simply averages based on the limited data available and for the cohorts tested. As always, YMMV.

The formula by Peronnet et al is the least aggressive at reducing the estimate of a cyclist’s power at higher altitudes, and that may be due to various not insignificant assumptions used in calculating each rider’s power outputs.

OK, so now we have estimates of both the physics upside and the physiological downside of altitude, What happens when we merge the two?

Well if I recreate the chart showing the physics, and overlay on that the curve showing power output as a function of altitude, this is what we get if we examine a rider capable of sustaining 51km/h at sea level:

Let me explain how to interpret the chart.

First of all, the vertical axis scale has been changed for clarity – the slightly curved coloured lines still represent the power to drag ratio required to attain a given speed at various altitudes.

So let's examine the case for a rider capable of sustaining 51km/h at sea level.

The thick orange line represents the power to drag required to sustain 51km/h. At sea level that's ~1,800 W/m^2 (Red circle 1). The exact value depends on a few other assumptions of course, so let's just use that as our "baseline" W/m^2 value.

Now if we apply the Bassett et al formula for power reduction for an altitude-acclimatised athlete, then their baseline sea level power (and with it their power to aero drag ratio) falls with increasing altitude. This drop in sustainable power with increasing altitude is indicated by the black dotted line.
We can see the power to drag ratio resulting from the physiological impact of altitude (the dotted black line) doesn't fall as quickly as the power to drag ratio required to sustain 51km/h (the thick orange line).

If you trace the black dotted line from left to right, we can see that at Red Circle 2, the power to drag ratio crosses the line marked 52km/h at an altitude of ~700 metres. Then as you trace the dotted line further to the right, we can see it cross the 53km/h line at ~1,500 metres. Tracing the line to the right hand edge of the chart out to 3,000 metres altitude, we can see it doesn't quite reach the 54km/h line, falling a little short at 53.9km/h. So for this altitude-acclimatised athlete, they can gain an extra 2km on their hour record simply by choosing to ride at an altitude of 1,500 metres.

OK, so what happens if the athlete is not acclimatised to altitude?

This time the non altitude-acclimatised power line is indicated by the lower black dashed line. It starts at 1,800 W/m^2 at sea level indicated at Blue circle 1, but as we trace that line to the right, it falls away more quickly than for the altitude-acclimatised athlete, crossing the 52km/h line at ~1,000 metres altitude (Blue circle 2) and not reaching the 53km/h line by the time the athlete is at 3,000 metres, where in this case the athlete would be estimated to achieve a speed of ~ 52.9km/h (Blue circle 3).

So while the acclimated athlete can improve their speed by 1km/h by going from sea level to 700 metres, and increase speed by 2km/h by going up to 1,500 metres, to achieve the same speed gains the non-acclimated athlete would need to ride at an altitude of 1,000 metres and would not be able to attain a 2km/h speed gain even at 3,000 metres.

We can see that as the altitude increases, the extra speed gains begin to diminish, and there are risks in going too high, especially if you are not acclimated, or experience an above average decline in power with altitude.
Conversely, if you are well acclimated and/or have a below average decline in power with altitude, then there are benefits in going higher if maximising speed is your primary objective.

Any rider considering an hour record would do well to consider the opportunity presented by tracks located at altitude. Of course costs, logistics, regulations all factor into the choice of venue, and how much time a rider may need to acclimate to altitude, and their individual response to altitude.

If a sea level based rider were considering a fly-in / fly-out attempt without much acclimation time, then I'd suggest choosing a good track that is not too high, as the risks of a larger than expected power decline increase significantly, and the potential speed gains diminish as well increasing complexity of execution as nailing pacing gets trickier. Of course the more experience a rider has with altitude and its impact on their performance, the more confident they can be with predicting an ideal location.

So what tracks are there at altitude?

Indoor laminated wooden 250m tracks at altitude include:
  • Aguascalientes, Mexico: 1,887 metres above sea level
  • Guadalajara, Mexico: 1,550 metres above sea level
  • Aigle, Switzerland: 415 metres above sea level
  • Astana, Kazakstan: 349 metres above sea level
  • Grenchen, Switzerland: 340 metres above sea level
There are track at much higher altitudes, but they are 333 metre outdoor tracks with concrete surfaces:
  • La Paz, Bolivia: 3,340m
  • Cochabamba, Bolivia: 2,571m
  • Arequipa, Peri: 2,295m
  • Mexico City, Mexico: 2,260m
Of the above listed tracks, Aguascalientes is a venue well worth considering. Eddy Merckx's October 1972 hour record of course was set in Mexico City, as were Francesco Moser's two hour records in January 1984. Most hour records since then have been set at or near sea level, with the recent rejigged rule records set by Jen's Voigt and Matthias Brändle at the Aigle and Grenchen tracks in Switzerland respectively.

So what's actually possible by the bigger guns of the sport. e.g. Wiggins, Martin, Bobridge and company?

I'll save that analysis for a future post, as well as a look at generating a formula to estimate the range of potential speed gains as a function of altitude, given an estimated sea level performance.

Read More......

Thursday, December 11, 2014

The sum of the parts II

Warning. Bike geek talk follows involving maths and physics, although I'll spare you the calculations and just provide the charts summarising the outcomes.

In this February 2013 post "The sum of the parts", I discussed the relative importance of wheel mass and wheel aerodynamics during accelerations using a technique known as forward integration, and based on the mathematical model of road cycling power by Martin et al.

As I said back then, aero and weight are not the only wheel performance factors to consider, but these are the two I am going to examine here, since  they are the two factors most often conflated with respect to their relative importance. Read my earlier post for a long list of other factors to consider.

In that item I made some assumptions about the scenarios being compared and said I would return one day with an updated version using better assumptions applied to the forward integration model.

I’ve now had the chance to do the revised modelling, and I was prompted by some recent forum postings once again promulgating the wheel rotating mass is really important myth, since it’s an often misunderstood concept.

So I’m going to run some numbers through the model for three scenarios:

  1. A standing start acceleration lasting 10-seconds on flat windless road,
  2. A rolling start acceleration lasting 10-seconds from an initial speed of 30km/h
  3. A rolling start acceleration lasting 10-seconds from an initial speed of 15km/h and heading up a 4% gradient

For the first I will examine the impact of adding mass to the wheel rims, and then for each when adding that extra rim mass comes with an aerodynamic benefit.

This time I have added extra features to the model to improve its realism. These are:

i. Including calculation of the rotational kinetic energy of the wheel in the model. Previously I had simply overstated the wheel mass difference as a (not unreasonable) means to compensate for not including a calculation of changes in rotational kinetic energy.

ii. Using a power curve in each scenario that is more realistic. In the original model I used a flat 1000W power curve, but of course no one generates power in such a manner. So this time I used power curves for a standing start and a rolling acceleration based on sample accelerations from my own (post amputation) data.

Rotational Kinetic Energy demand of accelerating wheels is small

The amount of additional energy or power required to accelerate a heavier rim at the same rate is very small, but nonetheless, I added it to the model for the sake of completeness. In this revised model I only considered the case of adding extra mass at the rim, since this is the “worst case scenario” for adding wheel mass, and happens to also be easier to calculate the moment of inertia.

If you want to really examine the difference between two wheelsets, you'd need to know the moment of inertia of each wheel, but it will always be less than if all the extra mass is added at the rim.

So just how much is the difference in energy demand from adding extra mass at the rim?

Well if you consider a 5-second long acceleration of an 80kg road bike + rider on a flat windless road from 30km/h to 40km/h, it requires a 5-second average power of over 700W. Adding 250 grams to the wheel's rims will require a whopping extra 2.7 watts to attain the same rate of acceleration, of which half is the additional power required for the translational (linear) acceleration component, and half for the wheel's rotational acceleration component.

Like I said, the extra energy demand to accelerate additional rim mass is not much. But every little bit counts. Sum of the parts.

Standing Starts

So let’s begin with the standing start scenario (i.e. an acceleration from 0 km/h).

How much does adding 250g to the rim affect acceleration?

Well here are the model assumptions comparing each set up:

Below is shown the respective speed curves if we apply a standing start power curve typical for me, i.e. power rises quickly to a little over 1000W after 2.5-seconds, holds near that level for about 5-seconds, and begins dropping away after ~8-seconds as neuromuscular fatigue sets in. For reference, the 10-second average power as shown is 924W. You can click on the image to see a larger version.

The difference in speed curves for each scenario is almost impossible to discern - there are actually two speed curves but they overlap very closely, hence appear as only one.

So in order to assess the differences, I plot the difference in cumulative distance travelled at each moment in time. In other words, this chart plots how far ahead or behind the second set up (heavier rim) is after X seconds.

We can see the addition of extra mass at the rim reduces the acceleration slightly and after 10-seconds there’s a loss of 22cm compared with using the lighter rim. Cool, let's all rush out to get lighter wheels. Well, just hang on a minute...

Heavier but with better aerodynamics

As mentioned in my previous post, I’ve already shown a drop in CdA of 0.023m^2 in low yaw conditions between using a set of low profile spoked wheels, and a deep section aerodynamic wheelset. But you don’t need to believe me, there is plenty of wheel test data in the public domain, for example the one done many years back by Roues Artisanales showing the power absorbed for various wheels. For good aero wheels, the advantage increases significantly in cross winds. There are many other wheel aero tests available, and Jack Mott of Aeroweenie has put together a neat list of such datasets.

So what happens if that extra 250 grams of rim mass comes with an aerodynamic bonus reducing CdA by 0.023m^2?

Since the plot of speeds still shows very little discernible difference, and I’m applying the same power curve, I’ll go straight to the cumulative distance difference chart:

Initially the lighter wheel takes an advantage and gradually pulls ahead of its heavier but more aerodynamic rival, gaining a maximum advantage of 5cm after 5-seconds. However, as the acceleration progresses, the bike/rider with the heavier but more aero wheel begins to catch up, draws level and passes the light wheel rider after 8-seconds and finishes the 10-second effort with an advantage of 11cm.

So, not a lot in it, but remember that this is a standing start scenario, which is the quickest acceleration scenario there is on a bike, and where the impact of wheel rim mass has the greatest (albeit minimal) impact on performance.

In my previous post using simpler model assumptions I said that the lighter but less aero wheel set was good for a standing sprint of up to six seconds. Well with a power curve that’s a little more realistic (for me), that advantage extends to all of eight seconds. Beyond then and it won’t matter, the bike/rider with the heavier but more aero wheel will pull away. And the greater your standing power curve, the earlier the advantage tips to the aero wheel.

Rolling accelerations

OK, so what about accelerations from a rolling start? Well you should be able to guess the outcome of this one before reading on.

Here are the power and speed curves for a rolling start sprint effort.

In this case you’ll note that the peak power is higher than for the standing start, closer to 1250W occuring again after about 2.5 seconds (and incidentally sees me add about 1000W above the baseline effort of ~250W), but thereafter drops away consistently. This sort of power curve is normal for me from a rolling start as I can get pedal speed higher and more rapidly but I also experience a quicker decline from that peak than in the standing start. 10-second average power in this scenario is 989W (about 65W higher than the standing start).

Here’s is the cumulative distance difference:

The bike/rider with the heavier but more aero wheel pulls ahead as soon as they start their sprint and never looks back, ending up with a 58cm advantage after 10 seconds, or nearly a full wheel ahead.

Now of course the assumption with these comparisons is that all other properties of the wheels are the same, even so my original conclusion stands, even in races with hard accelerations.

The model can be run with anyone’s individual power curve, mass, CdA and Crr assumptions, as well as considering other factors such as gradient and wind.

Hotdog crit anyone? Sprint up a hill?

Let’s say we have the same bike, rider and wheel sets as above but this time the acceleration begins from only 15km/h and goes up a 4% gradient. Nasty.

Here the lighter wheel gains a maximum advantage of just under 2cm after 3.8 seconds, thereafter the heavier aero rim catches back up after 5.7 seconds and ends the 10 seconds sprint 20cm ahead.

If the finish line were 90-100m or more from the turn, I sure know which wheel I’d prefer to be using. If it was only 45 metres though, well it’d sure be a tight race and you'd need a high speed finish line camera to pick the winner!

Keep it steady son!

OK so the point of all this was to demonstrate the relative unimportance of wheel rim mass and why aerodynamics matters even when accelerating, as dynamic scenarios are somewhat harder to calculate than steady state cycling scenarios such as time trialling, or hill climbing where accelerations are very small and changes in rotational kinetic energy are zero, or so tiny as to be completely negligible.

For steady state cycling, well the heavier but more aero wheel still wins in just about every scenario.

For the rider in this set of examples, at 300W the bike/rider with the heavier but more aero wheel still climbs faster on gradients of up to 8%, and of course will descend more quickly as well. At 9% gradient it's line ball and once you go steeper than that, well the lighter rim is quicker.

Read More......

Tuesday, October 14, 2014

Tinkov's Triple Tour Challenge: 10 Fun Facts

OK, so Oleg Tinkov has made an offer and it's got people talking. Which is probably his main aim, he's like that, never shy of a bit of entertaining nonsense or stirring the pot with ideas from outside the cycling box.

But I'm not so sure it's an offer too good to refuse.

So unless you've not kept an eye on any cycling news media channel over past week, then you'll no doubt have seen news of the challenge.

Here's the link:
Tinkov offers a €1 million to Contador and his Grand Tour rivals

In a nutshell, this is Tinkov's offer as quoted in the above article:

"If Quintana, Froome, Nibali and Contador all agree to ride all three Grand Tours, I'll get Tinkoff Bank to put up €1 million. They can have €250,000 each as an extra incentive. I think it's a good idea,"

Personally I just don't see it happening, simply because the risk to a rider's peak performance is too great and the proposed reward too little to compensate for throwing away the prize money and sponsorship attainable from a GT victory, especially a Tour de France victory. I'm just not convinced on the ROI.

Others have written about it and I don't propose at this time to add much to those discussions. For a couple of perspectives, see Inrng's comments about the practicality and marketing, and Science of Sport's take on the (not unsubstantial) physiological consideration:

inrng: Oleg Tinkov’s Indecent Proposal

The Science of Sport: Tinkov’s 3 Grand Tour challenge: Physiological, or folly?

Instead I thought I'd list some fun facts about the history of riders who have completed all three Grand Tours in the same year. Remember that the Vuelta a España only began in 1935, compared with 1909 for the Giro d'Italia and 1903 for the Tour de France. So we about talking about 70 years of all three grand tours, however due to various wars and a calendar gap, in 12 of those years not all three grand tours were contested.

So here are 10 fun facts about riders who have completed all three grand tours in the same year:

Only 32 riders have ever completed all three Grand Tours in a season (the same year).

The completion of all three Grand Tours in same season has only been been done 41 times.

Marino Lejarreta (ESP) did it four times between 1987 and 1991.

Adam Hansen (AUS) has completed 10 consecutive Grand Tours, the most by any individual. The first of this remarkable feat being the 2011 Vuelta and since the last was the 2014 Vuelta, he can extend that record in 2015 if he completes the Giro d'Italia.

Only one rider ever has won a Grand Tour and completed all three Grand Tours in one season. Gastone Nencini (ITA) won the 1957 Giro.

Podiums are rare from riders who complete all three Grand Tours. Including Nencini, only five riders have ever managed that feat.

Others podium finishers who also completed all three Grand Tours in the same year include:
Marzio Bruseghin (ITA) 3rd Giro 2008;
Marino Lejarreta (ITA) 3rd Vuelta 1991;
Bernardo Ruiz (ESP) 3rd Vuelta 1957;
Raphael Geminiani (FRA) 3rd Vuelta 1955

No rider ever has won or placed on podium at the Tour de France and completed all three Grand Tours in the same year.

The nearest to completing that feat was Carlos Sastre (ESP) with 4th place Tour de France 2006.

Only two riders have completed all three Grand Tours in a season and finished top 10 in each: Geminiani in 1955 and Nencini in 1957.

Read More......

Monday, October 13, 2014

Alpe d'Huez: TDF Fastest Ascent Times 1982-2013

In this June 2013 post I outlined the average speed of the five fastest times up Alpe d'Huez each year during the Tour de France since 1982.

The data was sourced from the posts by Ammatti Pyoraily on this Finnish forum. Thanks to him we have lots of data on times to compare over many years.

At the time of writing I hadn't the data for the 2013 ascent (since TDF is in July each year), so here is an updated chart for reference:

Here are the top 5 from 2013:

The Tour visits Alpe d'Huez again in 2015.

Read More......

Sunday, October 12, 2014

Power Meter usage still on the rise at Kona

Last year in this post I put together a chart showing the trends in power meter usage at the Kona World Ironman Championships since 2009.

Lava Magazine have once again done a complete bike and equipment count for Kona 2014, and I've been looking at the power meter part of that count. The data I have is preliminary as posted by Brad Culp of Lava Magazine. I'll post the online link with the count data when available.

2014 Kona IM Bike Count

Here's an updated chart and table for the six years from 2009 to 2014. Just click on the image to see a larger version.

In brief, we can see there has been a continuation of the strong trend in use of power meters, with 45% of all bikes now fitted with a power meter.

The two long established brands, SRM and Powertap, have fallen away a little in absolute numbers as well as total share dropping, while Quarq usage has grown again and it remains the dominant power meter brand for Kona IM athletes with more than double the usage of the next most popular brand, SRM.

Most of  the growth in total power meter usage is attributed to the use of newer power meter brands, with Power2Max, Garmin Vector and Stages being prominent in increasing the overall size of the power meter pie.

Speaking of pies, here is the 2014 breakdown in pie form:

It's interesting to note how evenly split the major power meter brands are.

What will 2015 show? I guess we'll see the number of bikes with power meters out numbering those without for the first time.

Read More......

Sunday, September 28, 2014

W/m^2, Altitude and the Hour Record

More hour record stuff to follow on from the item on Jens Voigt's hour ride.

This time to look at the physics impact of increasing altitude. I'll layer on top of this the physiological impact in a future post.

tldr version, click on this chart:

In brief:
- for a given W/m^2, you'll go faster as altitude increases
- for a given speed, the W/m^2 required reduces with altitude
- for a given altitude, to go faster the W/m^2 required increases

Now the long version:

There are two major factors which determine the speed a rider can maintain on flat terrain such as a velodrome, that being their power output and the air resistance. Or put another way, these are the primary energy supply and demand factors. There are other smaller energy factors as well (mostly on the demand side) but power output and air resistance are by far the most important when it comes to riding an hour record attempt on velodromes (or any race of individual speed on flatter terrain).

Energy Supply

What power output one can sustain for an hour is a function of several underlying factors that I discuss in this post. We influence that primarily through training, and of course to a large extent it depends upon the genetic gifts we are blessed with*.

There is of course also the physiological impact of altitude, as the partial pressure of oxygen reduces with increasing altitude, and as a result, so reduces the power we are able to maximally sustain aerobically (with oxygen). How much reduction in power occurs with altitude is individually variable, and you can acclimate to some extent as well, but there is no denying that once altitude starts getting high, ability to generate power definitely falls away.

I go through some of this in this post on altitude training, and I will be returning to this and its impact on hour records in a future post.

Energy Demand

For hour records on a velodrome, air resistance accounts for more than 90% of the total energy demand factors. In the case of  indoor velodromes and speeds in the 50-56km/h range, it's of the order of 92-93% of the total energy demand, with the balance mostly being rolling resistance and other frictional energy losses, and a tiny fraction in kinetic energy changes. This dominance of air resistance in the energy demand is why there is such a solid relationship between speed and the ratio of power to aerodynamic drag.

Air resistance & CdA

Air resistance on a cyclist is a function of several factors, being:
- the bike and rider's coefficient of air drag (Cd),
- their effective frontal area (A),
- the speed they are travelling at,
- the speed and relative direction of any wind, and
- the density of the air.

The coefficient of drag (Cd) and frontal area (A) multiply together to give us a measure of a rider's air resistance property - CdA. A lower CdA means you can go faster for the same power, or less power is required to sustain the same speed.

CdA is something a rider can change through bike positional and equipment choices (e.g. using an aerodynamic tuck position reduces your CdA compared with sitting more upright, or using deep section wheels with fewer spokes lowers CdA compared with using shallow box section rims with lots of spokes).

So to ride faster on an indoor velodrome where there is no tail or head wind to aid or hinder, you'll need to either:
- increase your power output, or
- reduce your CdA, or
- reduce the density of air you are riding through.

Or of course some net combination of all three that results in more speed.

It is possible that one can produce less power but have a significant reduction in air resistance factors such that the resulting speed is higher. For example, sometimes there is trade off between the advantage gained from use of an aerodynamic position on a bike, even though there may be a sacrifice of some power output due to the impact the aggressive bike position has on a rider's bio-mechanical effectiveness.

It all boils down to W/m^2

Robert Chung some years ago published a nice chart that shows the equivalency of speed on flat terrain with the ratio of power to CdA:

What we can see in this chart is how well Power/CdA can help estimate speed on flat road terrain over a wide range of power outputs and CdA values. Of course it's not a perfect correlation, as you can attain a slightly higher speed with the same W/m^2 as the power (and CdA) increases. So even if you share the same W/m^2 as another rider, the rider that has more absolute power will still be ever so slightly faster.

Not by much though. As an example, if we compared two riders on a low rolling resistance velodrome, one with 400W and another with 10% more power (440W) and both had the same power to CdA ratio of 1700W/m^2, then the more powerful rider will only be ~0.1km/h or 0.2% faster (all else equal). Like I said, there's not much in it.

I also showed this in the chart from my previous post on the Jens Voigt hour ride, where estimating his W/m^2 with reasonable precision is much easier than his absolute power. If the power was lower, so the W/m^2 must be a little higher, but not by much. Over a 100W (25-30%) range of possible power outputs, the W/m^2 required to attain the same speed varies by only 2%.

So even if we consider a range of power outputs typical for elite riders of the calibre likely to attempt an hour record ride, the W/m^2 ratio required for a given speed on a given velodrome will be within a pretty tight range.

Air density

Air density however isn't quite as easy for an individual to control, as it is largely a function of environmental conditions, in particular:
- air temperature,
- barometric pressure, and
- altitude.

Air density drops with an increase in temperature and altitude, and with a reduction in barometric pressure. Humidity also affects air density, but only by a very small amount (humid air is marginally less dense than dry air). So while a rider cannot control the atmospheric barometric pressure, they can choose a velodrome with a temperature control system, or one that will likely be warm, as well as choose from a range of tracks that are at different altitudes.

Altitude and its impact on speed

So given all that, I thought I'd look at how the combined effect of the power and aero drag values required to ride at certain speeds varies with altitude. As is typical of me, I've summarised this in a chart shown below. As usual, click on the image to see a larger version.

It's not overly complex, but let me explain.

On the vertical axis is the ratio of power output to the coefficient of aero drag x frontal area (CdA). Power / CdA in units of watts per metres squared.

On the horizontal axis is altitude in metres.

Then I have plotted a series of slightly curves lines, one each for speed ranging in 1km/h increments from 47km/h to 56km/h, and another line for 56.375km/h, which is the speed Chris Boardman averaged for his hour record.

For the sake of comparison, I've fixed the air temperature, barometric pressure, bike + rider mass and rolling resistance to be constant values for each. I did a little variation of power, but not much, and as I have demonstrated, the impact is very small.

So if we look at any particular line, we can see how the W/m^2 required to sustain that speed reduces as altitude increases. And of course we can see that for any given W/m^2 the speed you can sustain varies with altitude.

e.g. let's take 1800W/m^2. At sea level, the 1800W/m^2 line crosses the 51km/h line. As you trace horizontally from left to right, the 1800W/m^2 crosses the speed lines roughly as follows:
51km/h @ sea level
52km/h @ ~500m altitude
53km/h @ ~1000m
54km/h @ ~1450m
55km/h @ ~1950m
56km/h @ ~2450m

So naturally there is interest in using tracks at higher altitudes in order to ride faster and set records.

Now of course different tracks have variable quality surfaces, and so the assumption of rolling resistance being equal at all tracks is not valid, so any comparison of actual tracks should also consider impact of changes to coefficient of rolling resistance (Crr). Even so, since Crr accounts for only ~ 5-6% of the total energy demand, then track smoothness, while a factor, is more important when considering tracks at similar altitudes.

But what about power output at altitude?

Well of course there is a trade off between the speed benefit of lower air density at increasing altitudes, and the reduction in a rider's power output as partial pressure of O2 falls.

Hence as altitude increases, while a rider's CdA will not change, their power output will fall and hence their W/m^2 will also fall accordingly. So the W/m^2 line for any individual won't be horizontal, but rather trend downwards from left to right.

How quickly an individual's W/m^2 line drops away with altitude then determines the real speed impact of altitude.

So, what's the optimal altitude for an hour record?

I'm going to explore that in a future post (although I'm certainly not the first to have done so). So stay tuned.

* Pithy Power Proverb: "Choose your parents wisely".

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Tuesday, September 23, 2014

Hour record: Jens Voigt

Plenty has been written about Jen's Voigt's successful attempt at the new UCI hour record, set under the recently revised rules which permits the same bike set up currently used for the individual pursuit.

I thought I'd just add a chart to illustrate what sort of power and aerodynamic drag would be required to attain the result Jens achieved. The chart below summarises these key numbers and plots the CdA v power required, and shows the ratios of power to coefficient of drag area and power to body mass.

Where on that line Jens was, I don't know exactly, but it will be somewhere along there, or nearby. Click on the chart to open a larger version so you can see the numbers.

Jens' power to CdA ratio was in the range of
1715 - 1750 W/m^2.

Let me add some detail as to how the chart is derived, the assumptions used and key sensitivities. First the numbers we know.

Distance travelled and speed
Jen's official distance for the hour was 51.115km, which is calculated by the number of whole and partial laps completed x 250m per lap. Now we know from video that Jens did not always ride a perfect line around the track, and so his wheels actually travelled further than the official distance. Riding a good line is all part of the skill of track racing, so Jens likely cost himself some official distance.

So when calculating what speed Jens was actually doing, we'd need to know his actual wheel speed or distance per lap. However since air resistance acts mostly on where the centre of mass of the bike and rider is, which on a velodrome travels a distance less than that of the wheels, then we'd also need to factor in the lean angle of the bike and rider. Now I'm not going to attempt to do that. The data does actually exist as it was recorded by the Alphamantis Track Aero System which performs such calculations on the fly, but I don't have it.

In any case, I am going to assume that the extra distance travelled by Jens' wheels was cancelled out by the lean angle meaning Jens' centre of mass travelled about the same as the officially recorded distance. It's difficult without more data to be more precise than that, but it's a reasonable assumption.

Complicating the speed equation was Jens' pacing, which was somewhat variable, starting strongly, falling into a lull and then increasing somewhat in the final 10-15 minutes of the ride. So there would have been quite some variations in the power output during the ride. Of course the event starts from a an electronic gate that holds the rider, and there is some extra effort require to get up to speed which takes 10-15 seconds, so while it's a factor, it's a pretty small one in the the overall hour.

Here is a picture posted by Xavier Disley on his twitter account, showing lap by lap speeds, and when Jens got up out of saddle briefly:

In any case, I am going to work with the overall average speed of the rider as 51.115km/h.

Environmental conditions
Based on Weather Underground link the following conditions existed at the time of the ride:

Air pressure: 1012hPa
Humidity: 60%
Outdoors there was a light wind of 2-3km/h and no precipitation.

A spectator at the track reported the temperature indoors was 26C. Outside it was 20C with a maximum of 23C, so the reported indoor temperature is plausible and I'll go with that.

430m at Grenchen, Switzerland.

All of this provides an air density value of 1.114kg/m^3.

Rider and equipment mass
Trek reported via social media Jens' body weight to be 76kg. It may have been a little more but it's not a number that is particular critical to the calculations, as this is all about power and air drag.

Bike/kit mass - I'm going to assume ~ 8kg, again the calculations are not overly sensitive to this value.

As an example of this insensitivity, changing rider's mass by 5% only introduces a 0.3% error into the W/m^2 calculations.

Rolling resistance
I'm going to assume a coefficient of rolling resistance (Crr) of 0.0025, which is about typical for a quality set of track tyres on a quality wooden indoor velodrome. I did some calculations for Crr of 0.002 and 0.003, which is quite a broad range for such tracks and tyres and it only changes the power demand by approximately  +/- 1.5%. This is because rolling resistance accounts for less than 10% of the total energy demand for the event.

Power and coefficient of drag area
OK, so given all that, what power and aerodynamic numbers would be required to do what Jens did?

Well we can't really know what power Jens averaged for the effort unless Trek release the data, but what we can say is what his power to air drag ratio was. To ride that speed, it would be in the range of 1715 to 1750 W/m^2.

That's an average, as of course Jens' actual instantaneous CdA did vary as he changed position on the bike at times. He was mostly in his aero bars but was occasionally standing up on the pedals or making other adjustments.

This chart shows the line along which Jens likely falls somewhere. If his power was lower, then his CdA must have also been lower in order to maintain power to drag ratio in the range of ~ 1715-1750 W/m^2.

The chart also then shows what his power to body mass ratio would be (assuming 76kg), so we can see the wide range of power capability possible to attain such a speeds. You don't need big power, if you are very slippery through the air.

Aero matters. A lot.

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Thursday, August 14, 2014

The Null Hypothesis

Recently a new aerodynamics cycling product came to my attention, and the company behind it makes some extraordinary claims about the speed improvement attainable. As Marcello Truzzi would say, extraordinary claims require extraordinary evidence. So let's have a look at one such claim.

The product is the Nullwinds Upper Wheel Fairings. Here's their weblink:

The idea is pretty simple, add some fairings that cover the upper part of the wheels and bingo, instant aerodynamic improvement. Well that's not so remarkable, it's pretty common to attain an aerodynamic improvement through use of fairings. It's also why such things are banned in competition for cyclists and triathletes, but that's not the issue here as Nullwinds is targeting this to non-competitive riders looking for a speed advantage.

OK, that's fine, we all could use a boost.

Just so it's clear, an aerodynamics improvement means the drag coefficient of the bike and rider is reduced so that you require less power to sustain the same speed, or for the same power you can ride faster. Nice.

So let's examine one such scenario as listed on their website as being a Strong Headwind Test:

NOVICE RIDER February 9, 2014 
1. Test Summary 
The best available data taken on February 9, 2014, indicates that the use of our Upper Wheel Fairings on a typical road bike with a novice rider under strong headwind conditions yields gains in average speed exceeding 20 percent (22.2 percent was recorded). (The full report is available for download.) Power measuring tests in severe headwinds were conducted on identical multi-speed road bikes configured with and without wheel fairings. A novice cyclist was the rider. Data was recorded using an i-bike Newton power meter. 

2. Implications The results confirm that the use of our Upper Wheel Fairings can dramatically increase headwind penetration speeds of a novice rider under strong headwind conditions. Gains exceeding 20 percent are possible.

So, Nullwinds claim a novice rider riding into a strong headwind will be able to achieve a speed gain of more than 20% by putting these fairings onto their bike.

Well to Nullwinds credit they have at least published some information in an attempt to back up their claims. They:
  • did some testing to attempt to demonstrate the effectiveness of their product (tick)
  • attempted to establish some testing controls (tick, but they were not so successful as we'll see later)
  • published data for some of those tests (tick - more detail in pdf here)
  • claimed some impressive results (hmmm, no tick)
Unfortunately, Nullwinds missed an important step before the final one, which was to examine their own data before making their extraordinary claim. So let's do that step for them.

The details of their testing protocols and measurements are outlined in the document and I won't repeat them here, just summarise: They used two identical bikes each with an ibike Newton bike computer/power meter as a data logger, one bike fitted with the Nullwinds Upper Wheel Fairings, and the other without. They asked a novice rider to ride into the wind over a designated section of pretty flat road, doing a run or runs on each of the bikes, and to keep their effort level about the same for each run.

All the bike/rider data was recorded by the ibike, charts are shown in the pdf document along with other information such as weather conditions, details about the venue and tests controls. I'll list all the important details below.

Wind: Headwind of 23mph (10.3 m/s)
An attempt was made to ride each bike in similar wind conditions, so I'm going to take their word for it. You can read details of how they managed that in the document. Whether this is the actual headwind faced by the rider is hard to know, they are relying on the ibike Newton to provide the data.

Power: 149.4W
They reported 149.4W for the rider on the non-faired bike. I'll crunch the numbers to see what reduction in CdA is required to attain the claimed speed improvement at the same power. I'll also come back to this, as the power output reported for the faired bike run was not the same as for the non-faired bike run. Power is of course being reported by an ibike Newton, so who knows how reliable the data really is, but nonetheless let's assume that's the actual power and check the numbers to see if it makes sense (turns out it does, more or less, if you believe the wind speed data).

CdA: 0.372m^2 (non-faired bike)
They report a coefficient of drag area of 0.372m^2 for the non-faired bike. I've no reason to question whether that's correct or not, it's a plausible number for a novice on a standard steel framed road bike. We are of course testing relative changes due to the fairing in any case, and we'll just have to assume the rider maintained the same or very similar position on the bike.

Crr: 0.0054
They report a coefficient of rolling resistance of 0.0054 and again I've no reason to suspect that's wildly wrong as it sounds plausible for road bike on a road. I will keep that constant (as they did).

Gradient: +0.29% (unfaired) and +0.55% (faired)
This one is tricky as they report a different average road slope for each test. +0.29% non-faired test and +0.55% for the faired bike test. While the test was conducted over the same 1.5-mile stretch of road, they chose slightly different 1-kilometre sections from each run's data to make the comparison. They did this to choose a section which provided the same average headwind speed.

Mass: 188lbs (85.3kg)
They report 188lbs. I don't know if that's bike + rider or just rider but I'll assume that's total mass, and there was no mass change between the rides. On flat terrain, the outcomes in terms of impact on speed are quite insensitive to changes in mass anyway.

Air density: 1.108kg/m^3
They report 70F (21.1C) and 1020hPa for their calculations, no humidity reported but weather report they provided shows that to be between ~30% and 50%. I'll use 40% (the air density calculation is very insensitive to changes in humidity anyway). They don't report elevation but the road used was right next to Fox Airfield in California and the airfield is reported to be at an elevation of 2351 feet (717 metres) above sea level. That gives an air density of 1.108kg/m^3.

So with those power and other assumptions, using the model by Martin et al, you'd expect a rider on an non-faired bike to attain a speed of 3.32m/s = 11.93 km/h = 7.42 mph

They reported an average speed on the non-faired bike run of 7.2mph. So on the whole, the numbers seem to be in the right ball park.

OK, so what improvement in aerodynamics, that is, what reduction in CdA would be required, all else the same, to attain the claimed speed increase of 22.2% (i.e. from 7.42 to 9.07 mph)?

The CdA required at same power would be 0.244m^2.
That's a reduction in CdA of nearly 0.13m^2, a 34% reduction!

That's the equivalent of removing all of the air drag of the entire bike and some of the rider!

Houston, we have a problem.

Now here's the kicker: the faired bike run reported average power of 202.9W, some 53.5W (+35.8%) more than during the non-faired bike run. Nullwinds also reported the rider's heart rate was 10% higher for the faired bike run than the non-faired bike run.

It's no wonder the rider went faster on the faired bike.
They simply rode harder.

So knowing that, what did Nullwinds report the faired bike CdA to be?

0.369m^2, a drop of only 0.003m^2 or just 0.8% less than for the non-faired bike.

It's a real marketing bugger when the actual size of your "benefit" is quite a bit less than the error in measurement, and doesn't sound anywhere nearly as impressive as a 20+% gain in speed.

Sorry Nullwinds, your claims of big speed improvement attainable as a result of using your Upper Wheel Fairings are not plausible. Unless perhaps there are secret stashes of EPO hidden behind them.

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Wednesday, June 18, 2014

Positioned for Speed

Last week I had the pleasure of co-delivering the first "Positioned for Speed" Course held in Australia, which is part of Retül University's growing list of international course offerings. Many thanks to Matt and Nick at Retül and Andy and the guys at Alphamantis for the opportunity, it was a lot of fun. Looking forward to doing more of them (if they'll have me back that is!).

The two day course was aimed at bike fitters and coaches primarily, and gave attendees an introduction to the theory of aerodynamics relevant to cycling, an understanding of how the theory applies to the practical considerations of bike fitting, what elements of aerodynamics we can influence and improve, how we quantify the impact to performance, as well a chance to design and conduct an aero testing session with a test subject.

I had fun explaining the theoretical aspects, then helping the participants understand and experience exactly how to translate these into actual testing scenarios, and using the Alphamantis track aero testing technology to measure the impact they have on a rider's performance.

We tested bike position options, equipment options (helmets and wheels), body shaping options while riding, and clothing options. Over the course of the session, incremental improvements in the rider's aerodynamics were identified, all while ensuring the rider's position was still bio-mechanically effective and comfortable for the rider when considering the events they are targeting.

Thought I'd share a few examples of comparison test results along the way. I can't say much about the rider, or the exact details of each options tested, but suffice to say they are targeting road time trials and track endurance events.

Put a lid on it

Aero helmets are known to give good aerodynamic benefit but which helmet is best for any individual is quite variable. In any case, the team immediately saw the sizeable benefit of one aero lid over the rider's existing standard "mass start" helmet. These were not the only options tested but just shown as a comparison example.

Putting that into perspective, at this rider's Function Threshold Power, that's a gain of more than 0.6km/h or 1.1 seconds per kilometre on flat road terrain. Some people will gain more speed and some less from an aero helmet, and no one helmet brand or model is the best choice for every rider. Some provide more speed gains than others.

The value of a good shrug

Next example is how you can gain speed by "shrugging" (or "turtling") such that you bring your head down and narrow your shoulders while riding in the TT position, but do so without compromising your power output. Sometimes riders learn to be able to do this for extended periods of time, but it's a technique mainly for shorter road TTs and individual pursuit, not so much for the Ironman athletes out there. The gains can be well worth it if you are able to hold onto a shrug for a while.

In this rider's case, they can increase road time trial speed by nearly 0.5km/h or gain nearly 0.9 seconds per km while they shrug. For some riders there are bike position set ups and helmets that enable the rider to shrug more easily or hold it for longer. Ideally you'd like to set up the bike such that the effect is a full time enhancement, however this is not always feasible, so being on the lookout for more free speed-gaining opportunities is worth a go.

Skinsuits. Choose wisely.

The final example I thought I'd share from the testing session was some skinsuit options. Here we can see the difference between three suits tested.

The best suit is about 0.4km/h or 0.6 seconds per km faster than the team issue suit at this rider's pursuit power on a track. That gives them a 25 metre lead over the slower suit by lap 12.

Overall we identified a 0.033m^2 reduction in this rider's coefficient of drag area, which is equivalent to a 35 watt power saving, or a little over 3 seconds per km or a speed gain of 1.7 km/h.

Talk about a winning margin.

Discussing track test routine with one of the course participants.

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Saturday, May 31, 2014

Swings and Roundabouts

Today I thought I'd look at how rider's relative abilities play out in Grand Tour stage racing. I was prompted to do so after watching this year's Giro d'Italia's Stage 19 mountain individual time trial.

It's no surprise that to be competitive on general classification (GC) in a grand tour requires, amongst a variety of traits, phenomenal physiological capabilities, and each rider has their strengths and weaknesses relative to their opponents. Clearly climbing ability is a key factor in success, but also a rider's ability in individual time trials (ITT) is also an important factor for success.

Success in the ITT, is more suited to those riders with the highest sustainable power to aerodynamic drag ratio (power to CdA, W/m^2), while climbing steeper gradients a rider's sustainable power to weight ratio (W/kg) is the dominant factor for success.

Some riders are a bit larger and have more power and are relatively more aerodynamically shaped and set up, while others have better power to weight ratios. If you can nearly equal your opponents in one discipline and beat them in the other, then you're well on your way to GC success.

The balance between these two key attributes does vary from race to race as grand tours vary the total number and distance of ITTs, and the number of mountain top finishes. While total metres climbed during a grand tour is a factor for overall fatigue levels, it's the summit finishes and individual time trials that account for the majority of time gained and lost between leading contenders, and these are the critical stages for GC riders.

As an example of this game of physical swings and roundabouts, note the differences in make up of race defining stages in recent editions of the Tour de France:

We can see that the 2012 TdF was far more suited to TT ability than it was for climbing ability, but this year is more weighted towards the climbers. Of course you can't badly falter in either discipline, but 2012 was never one for a pure lightweight climber.

In this 2014 edition of the Giro d'Italia, there have been three time trials: a team time trial, an individual time trial over flat/lumpy terrain and an individual time trial (MTT) up a mountain. There are also 5 high mountain summit finishes. So while ITT performance is important, it was always going to be a rider's climbing prowess and their power to weight ratio that dominated this Giro.

What is interesting with this race though was the inclusion of the MTT, giving us a chance to directly compare riders' flatter ITT and MTT performances.

Both the individual TTs arrived fairly deep into the contest, with the ITT on stage 12 and the MTT on stage 19. I thought it interesting to plot the relative performance of the top GC riders in each of the time trials (click on the pic to see a bigger version). I chose the top 25 on GC (after Stage 19's MTT) as these are the riders more likely to be actually competing on such stages, rather than holding back somewhat to save the legs as much as possible for other duties or race ambitions.

The dots represents each of the top 25 riders on GC after Giro Stage 19. The top 10 riders are highlighted with red dots and text.

The rider's time for the Stage 12 ITT is plotted on the horizontal axis versus their time in the Stage 19 MTT on the vertical axis. e.g. we can see Uran's time in the ITT was a bit over 57:30, and in the MTT a little over 1:07.

When plotted this way we can see whether riders fared relatively better in the flatter ITT, the MTT, or if they performed relatively similarly in each. Of course the further a rider's dot appears to the bottom left indicates faster TT times overall.

The further away the rider's dot is from the diagonal line indicates a dominance of either the ITT (upper left) or the MTT (lower right). If their dot appears close to the diagonal, it indicates their relative performance for the two time trials was balanced.

Those in the upper/left side of the chart are more skewed to perform better in flatter ITTs, and hence their power to aero drag ratio is relatively better than their power to weight ratio, while those in the bottom right side of the chart performed relatively better in the MTT and so their power to weight ratio is relatively better than their power to aero drag ratio.

This Giro is clearly one for the climbers over the TT men given there is an MTT and 5 high mountain summit finishes, and only one (not totally flat) ITT and one TTT.

The balance between each of these two key attributes will play a factor in a rider's likely chances of success in any grand tour and which side of that balance is more dominant depends in large part to the make up of the critical stages. It also points out what element of performance a rider may need to look at in order to improve their overall standing relative to their opponents.

Looking at the chart, there are some riders that would do well to refine their ITT performance, something that's more possible to address with a focus on aerodynamics. Losing weight and gaining power is much harder for a top Grand Tour GC rider to (legally) achieve.

Aru, Rolland and of course Quintana will be dominant riders for the future if they are able to improve their power to aero drag ratio. Of these three, I'd say Rolland and Quintana have the most to gain from aero refinements.

Pierre Rolland
Fabio Aru
Nairo Quintana

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Thursday, February 06, 2014

Sands through the hour glass

A couple of charts for a bit of fun.

Recently we read about 102 year old Frenchman, Robert Marchand, who last week set a new best distance of 26.925 kilometres for the hour at the new velodrome in Saint-Quentin-en-Yvelines, France. It was widely reported in many journals and blogs. Here's a link to item but a quick Google search will show lots of reports. Chapeau to Robert!

Having been involved with a number of age group category hour records, I thought I'd chart all the current records, including the two outlier points of Marchand, and Chris Boardman.

Here they are in table form with details as at February 2014:

I included Chris Boardman's record in the men's list for reference, and because it was set using the then pursuit bike set up rules which are the rules that masters age category records are run under (although those rules have been modified somewhat since Boardman's phenomenal ride).

I note however that our famous centenarian used a regular (Merckx-style) bike set up, but let's not be overly concerned with that. Being upright, let alone riding is super stuff at that age!

Here are all the age group record holders plotted showing age and distance covered in the hour. Click on the pic to see a larger version.

For some added fun I drew a line linking Boardman's and Marchand's records. The slope is just a touch under 400 metres less per year of age. With the exception of the younger masters age categories up to about 40/45 years, the men's records seem to roughly follow that level of performance decline with age, perhaps with a decline closer to a metre per day of age.

There are fewer masters women's records and none past 66 years old, so it's harder to say if their rate of performance decline is comparable, so while eye-balling suggests the rate of decline is less severe than for the men, I'm not sure I'd draw too many conclusions from these data. There are so many variables, and as the ageless TV soap used to tell us... like sands through the hour glass, so are the days of our lives.

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