Time To Fatigue

Tiny changes in power can change your career.

The Lads.

This is the 3rd post in our "Maths of Cycling" series. Click here to view our last post. Today we are going to look at how to make big riders climb faster.

If you are a big guy and want to climb fast, first you need understand why climbing is hard for you.

The Real Why

The reason you get out-climbed, as we saw in our last blog, is not because smaller riders weigh less or produce more power, it's because their power output doesn't change much from when they're on the flat vs. up a climb.

Having a high drag compared to their body weight forces small riders to ride at a higher Watts per Kg (W/Kg) on the flat to keep up. This handicap feels normal to them and over time it makes riding everywhere at a higher W/kg easy. W/Kg is all that counts on a climb and it means they don't even feel it when they start to drop you on the hills.

Rider A has a 0.27w/kg advantage with no extra effort - that's more than 8% climbing performance.

If you want to improve your climbing you need to start by reducing the difference between your power on the flats and your power on the climb. It's actually pretty simple to do this. We need to increase the drag of the bigger rider until his power on the flat matches the power needed to keep up on the climb.

In this situation, Rider B needs to add 23w at 32km/h or 0.05 CdA. You could try making your brakes rub or even letting your tires down, however this resistance from the flat will not go away for the climb and will make your ride even worse. (CdA mode on the AIRhub will automatically reduce resistance on the climbs to perfectly match the situation.)

Rider B (red) added 0.05CdA in the AIRhub app and now rides with the same Watts per kilo

It is as simple as that. The effort level of the large rider no longer needs to increase to keep up with the small athlete when the road tilts up.

Failing to add resistance to your ride basically leaves you with no hope of ever keeping up. 

This is Why:

To climb with the same effort level, both riders need to have the same threshold. Let's make this 4w/kg. Next, we need to look at a Mean Maximal Power curve. A Mean Maximal Power curve uses your threshold and ride data to tell you how long you can hold a power output for before you fatigue. With some tricky maths we can work out exactly when each rider will fatigue. The calculations we used are at the bottom.

At 32km/h on the flat, Rider A will ride for 20hrs until fatigue (remember he has to ride everywhere with a higher Watts per kilo). Rider B can ride for 46hrs until he fatigues because his drag (CdA) per kilo is less, making his ride is easier. To achieve the same training load Rider B must ride an extra 26hours per week. 

If Rider B is lazy and only does 20hrs per week like his friend, the orange curve will drop vertically and wipe 10.7% out of his threshold. The orange curve determines the maximum power you can ride for a duration.

Without the extra training hours Rider B will de-train causing him to lose his threshold and performance on the climb. The smaller rider will go back to kicking his ass without even feeling it.

To obtain that few extra percent in threshold power that was lost, a massive increase in training duration is required. This is an increase in training duration that no one can match, and is why many people struggle to increase their threshold.

If you want to climb better you need to start by reducing the difference in your power between the flats and climbs. 

Next time, we will analyse an AIRHub optimised training ride between a Tour de France rider and a club rider.


For the first section we used the maths from the last blog.

The cda values remain the same, however for the intervention situation we added 0.05 CdA to boost Rider B's flat riding power/resistance.

To calculate power for a speed we use:

Power = Cda*0.5*Air Density*Velocity^3  + (0.005*(Mass+Bike mass)*9.8*Velocity) 

we used 1.182 for air density

TIme to Fatigue

To find Power, Threshold and Time to Fatigue we use the following equations   

Excel plot of Power and Time gives us trend line equation.

The trend line equation will tell us how quickly your power drops off against time.

The data above is a 1 year sample from a national level racing cyclist. Data can be pulled from Todays Plan or Training Peaks mean maximal power charts.

The trendline equation above is y=382.06x ^(-0.121)

Y = Required_Power (or power output.)

382.06 = athlete threshold calculated by the trend line.

(-0.121) = the exponent which tells us how fast power will drop off against time.

X = time in hours

That gives us; Required_Power = (Threshold) * (Time ^(exponent))

For our example we need to re arrange the equation to find "Time" 

Rearranging the equation we have; Time =(Required_Power/Threshold)^(1/exponent)

If we really want to get Tricky we can find the Threshold power of Rider B if he were Lazy and didnt complete his 46hrs training per week. In this example we want to find "Threshold"

We rearrange to: Threshold = Required_Power/(Time^(exponent))

Giving us Threshold = 221/(20^(-0.121)

Threshold = ~317w or 3.6wkg - 10%less