# Bicycle Velocity Prediction

This article and the associated programs were written at the dawn of the Internet age, between 1994 and 1996.

Have you ever wondered how much faster you would go if you spent \$5 on a pair of titanium water bottle cage screws? How much faster you would go if you lost ten pounds? Whether you should go anaerobic up that short hill? If that almost imperceptible headwind could be responsible for your slow speed (and not last night's chili)?

I ran across a computer program at a bicycling archive site (draco.acs.uci.edu), written by Ken Roberts, Columbia University, that could calculate the power required to maintain a speed under various conditions. I thought it would be more useful to predict speed from power, and offer the following programs that you can download or use directly from your browser:

The calculations consider the three forces felt by the rider: aerodynamic resistance, rolling resistance, and gravity. There is also some loss in the transfer of energy from the feet to the road, about 5%.

Multiplying weight times the grade (vertical feet divided by distance traveled) resolves the gravitational force in the direction of travel. Multiplying weight times a rolling resistance factor (about 0.004 for thin road tires) gives the rolling resistance.

Aerodynamic resistance is somewhat more complicated. The most significant aspect is that it depends on the square of your air speed - the sum of travel speed and headwind. You multiply this value by a coefficient that depends on an aerodynamic shape factor, area and air density.

Power is the product of force times velocity. So, you add the three forces above, and multiply by speed. If you convert everything properly to metric units, you get watts.

I wanted to predict speed from power, however. The power calculation is a cubic expression, meaning it is not easily re-arranged to calculate velocity directly from power. The programs use Newton's method to solve this expression, a fancy trial-and-error method.

One of the spreadsheet programs can represent a course with multiple legs. Each column calculates speed for certain conditions over a certain distance. The collection of columns can represent the segments of a ride. A simple out-and-back can be represented by two columns. A "Totals" column gives the total time, average speed, average power output, and calories consumed. There are interesting things to learn even in the simple case of an out and back climb. The answer to the question at the start of this page about charging up a hill is: yes, do it.

These calculations seem accurate. For example, I know I can ride a Lifecycle more or less continuously at a setting where it tells me I am consuming 800 calories per hour. This is what the calculation says I burn when I go 19.2 mph on the hoods, level, no wind. This is about right.

Let's answer some simple questions. First, why are even small headwinds so unpleasant? The following graph shows the effect on a rider putting out 150 watts for two positions, on the hoods (lower) and using an aerobar (upper):

A headwind of 5 mph reduces speed by nearly 3 mph. This is the equivalent of a 0.73% grade. You would have to increase power from 150 to 215 watts to compensate. And don't we all try, slaves to our cyclocomputers?

Here is the effect of grade on speed, for the same two positions (hoods and aerobar), at 150 watts:

I thought it would be interesting to see how weight would influence these curves. If I lost 10 lbs (about 5%), I would be able to go about 5% faster on the steepest hills, 0.4% faster on the level, and about 2% slower on the downhills.

Over a simulated 20-mile closed-circuit ride with a variety of grades, a 10-lb difference produced a 33 second difference. This may or may not seem significant in the context of a time trial. On the other hand, there were two hills on this simulated route where the heavier rider falls back 14 seconds. That is, about 200 feet back and well-dropped. A two-lb difference that you can buy at a bike shop for \$500 amounts to only 7 seconds on this circuit, but again, a two-lb penalty can mean cresting a hill 50 feet behind your better-sponsored buddies.

Here is a chart that gives speed versus power for the same two positions again:

The advantage of the aero position seems overstated (it is more accepted that it is worth one or two mph). But for me, the aerobar position is not entirely comfortable, and I don't seem to be able to peddle as hard. (I think the ten pounds I need to lose has something to do with this.)

By the way, in the final stage of the 1989 Tour de France, Greg Lemond averaged 34 mph with a 5 mph tailwind. According to the calculations here, he was putting out 513 watts.

If you can do 17.2 mph when riding on the brake hoods on a road bike with thin high pressure tires, you can only expect to go 14.4 mph on a fat tire mountain bike. This results from the increase in rolling resistance from 0.004 to 0.012.

About the author: I was an engineer (PhD) at GE Aircraft Engines at the time I created these programs and wrote this article. That is, left-brained. Since then I've become a full-time photographer (mixed-brain). Because of that switch, and 12 years of neuron loss, I can no longer do calculus or respond well to technical questions. Hell, math was always a stretch for me - I was a metallurgist.

Curt Austin, Austin Image