Take 2 riders. Rider A is 68kg/180cm, rider B is 88kg/190cm.

Both riders are using the same, top of the line 6.8kg racing bike. Both have the same body fat, same performance level, all are equal in all other things.

Both riders are going to ride up a hill with a power output of 5 watts per kilo. Their average speed will be under 20kmh, so wind drag in this scenario is negligible. Instinctively, you would think that the lighter rider will win the race. We think this because we often see lighter riders winning mountain stages and we jump to the conclusion they have less weight to carry up the hill.

Back to our scenario; without jumping to conclusions lets look at the situation a bit closer.

Both athletes are riding at 5 w/kg, this means that the power output is equal to their weight. So being lighter should have no advantage. Considering this, it seems that both riders should cross the line at the same time. However, there is still one factor being left out.

The last consideration in this scenario is the ratio of the bike's weight to the riders weight. Both of the riders have the same 6.8kg bike, the UCI minimum weight. For rider A his bike now becomes a 10% dead weight over and above his body weight. Rider B, riding the same bike is only carrying 7.7% extra dead weight. The difference in bike to rider ratio presents the first mathematical difference between the cyclists.

The addition of the bike's weight into the equation shifts the watts per kilo output. Rider A's 10% dead weight compared with Rider B's 7.7%, gives Rider B a 2.3% advantage. This 2.3% advantage gives the heavier rider an additional 23m per kilometre. Over 10km, this will equal a 230m lead over Rider A.

So we've found the difference between the two riders, but it's in the wrong direction. This calculation is in contrast to our initial assumptions about the lighter rider. We've concluded that the heavier rider should have a 230m lead over 10km. Mathematically, the heavier rider *should *climb* 2.3%* faster.

The real world doesn't reflect the maths we've done. The cyclists we see crossing the line first in mountain stages are the lighter riders, not the Rider Bs of the peloton. If Rider B has the mathematical advantage, then Rider A must have some tricks up his sleeve.

Not sure about you, but Rider B really wants to know what they are.

*Please Note: Distances and advantage may vary depending on gradient and calculations used however the concept that Rider B should rider faster remains the evident.*

Same as the last blog post: Rider A is 180cm & 68kg, while Rider B is 190cm & 88kg. They are using the same, top of the line 6.8kg racing bike. Both athletes are riding along on the **flat**.

For this article we are going to keep things simple. We can calculate their power output, speed and estimate CdA with published calculations, so if you want the complex maths we've added them down below.

Each athlete is subject to a unique drag profile, also known as their Coefficient of drag (CdA). As a general rule, the more surface area facing the wind, the greater your CdA. On the flat, CdA is the most important factor and will decide how much power you need to put out for any given speed.

Different sized riders will have different drag profiles and different power requirements, this means that for a lighter rider, Rider A, to train at 32kmh he will have to output 189 watts(w). Rider B would have to output 221w to maintain the same pace.

To ride at 189w Rider A is putting out 2.78 watts per kilo (Wkg). Rider B riding at 221w, is putting out 2.51 Wkg to maintain the same pace.

Even though Rider B must push more watts, he has more muscle mass to do so, meaning his Wkg are lower and his flat rides are easier. This is part of the reason why we see big athletes winning flat races all the time.

This also means that due to Rider A's physique compared with Rider B's, he will be training 10.7% harder than his training partner.

Rider A is training harder than Rider B by 10.7% on every single ride they do, and very shortly he will have adapted to it. The best part about this for Rider A is that he may not even notice he is training harder; He is just riding with his training partner, and riding harder is something he has always lived with.

When it comes time to leave the flat and venture back to the hills it is the riders that have trained the hardest that will succeed. 10.7% extra training on every ride is going to come in handy.

If both riders ride the hill at the exact same power to weight as they were riding on the flat, 2.78 Wkg and 2.51 Wkg respectively, Rider A will climb **8.4% faster** (this includes the 2.3% advantage to the larger rider for the bike weight - from our **last post**).

Now, this is a picture we are more used to seeing.

Every time Rider A trains with a higher Wkg, he is going to become a better athlete than you.

Pretty sneaky if you ask me.

**Using the Height and Weight of the athlete we can Estimate their Cda using the following published equation.**

*Bassett et al. (Med Sci Sports Exerc 1999; 31:1665-1676):*

*Frontal area (m^2) = 0.0293 x height (m) x mass (kg)^0.425 + 0.0604*

*n=8; R^2 = 0.76; P = 0.05; S.E.E. = 0.009 m^2*

*Cd on hoods =1. Cd on Aerobars = 0.7*

**To calculate power for a speed we use:**

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

*we used 1.182 for air density*

**To find time on a slope use:**

*Time(s) = Distance / (Power / (Slope%* (Mass + Bike Mass) * 9.8))*

NB: For the hill climbing part of this equation we negated Cda as per the last blog. The reason i did this is for simplicity, allowing us to use what we learnt last time. Including Cda, the smaller rider's advantage reduces slightly.

]]>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 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.

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.)

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.

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.

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.

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

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