# The maths of flat riding

### The harder you train the more you will get beaten.

Last post we explored the mathematical inequality between riders of different physique. Finding that larger riders should climb faster than their smaller counterparts. In this post we'll investigate what actually happens to allow small riders to climb faster.

### Scenario

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.

### The silent enemy

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.

### Super sneaky training

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.

### The "hard-core" maths.

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.