# Power vs Velocity predictions

Power meters are a great tool to help you with training, as well as establishing proper pacing for races… I have found my PowerTap to be a great addition to my bike.

Of course once I starting looking at the data, I wanted to see if I could start to trend my bike power versus speed to identify a correlation.  I’ve looked at sites like:

Although I think these sites are okay for comparing positions (i.e. how much speed you can gain by going from drops to an aero bar), I’ve never found that they correlate well to my power versus speed values.  I started tracking some of my rides and plotting the data.
The power required to move a bike is a function of three main components:
• Air resistance – This is the aerodynamic drag associated with moving you and your bike through the wind.  I’ve read that about 70% of the drag is actually by your body, and about another 70% of your body drag is determined by your body position (get low and aero).
• Friction – This is the mechanical friction of your bike and tires.
• Slope – Gravity needs to be overcome to go up hill… and that takes watts.  Likewise, without applying any power to the pedals, your bike will accelerate downhill up to a point of terminal velocity (when the force of gravity is offset by the friction and air resistance).
With most of my rides being out and backs, as well as many local triathlons, I decided to simplify the power equation by eliminating the slope component.  Basically this assumes that if you go uphill on the way out and have to generate extra power to overcome the slope, you will get an equal and opposite offset by the benefit your get from gravity on the way back.
This leave the power equation as:
Power = Ca * V^3 + Cf * V * Wt
• Ca = coefficient of air resistance
• V = Velocity or speed
• Cf = coefficient of friction
• Wt = weight of rider plus bike
Plotting some of my rides and races, I came up with the following graph:
Although the data is not perfectly aligned, you can see a general trend in the numbers.  As you look at the trend line, any points to the left of the line are “less efficient” days while the points to the right would be better efficiency days.  In looking at the two data points on the left of the line near the bottom, these were days early this year when I was wearing tights and a coat… which probably negatively impacted my overall efficiency by increasing my wind resistance.
The challenge is then taking the equation above and fitting it to the data.  I did this by starting off with “typical” Ca & Cf factors (from this discussion) and altered them until I got a reasonable fit on my data.  What I found was that I had to decrease the wind resistance value and increase the mechanical resistance value to get a fit.  The plot of my actual and predicted values are shown below. I then took these values and plotted a model to project my average speed based on average power levels.  As I build power through training, this will give me some indication of the type of speed that you can average over a given race.  Understanding that there are a lot of variables, this will only be an average value… but for me it is much more accurate than the generic sites listed above.  The plot is shown below:
For reference, my Ca ended up being 0.12 and Cf was 0.053.  Note that the velocities need to be converted to m/s in the equation above for these particular coefficients.