Category Archives: Cycling

Analyzing Cycling Scenarios

Analyzing cycling scenarios

Let’s first put analyzing scenarios into context. We use Physical Cycling to do two things. The first is to understand cycling dynamics we we can improve our riding techniques. The second is to analyze particular cycling scenarios to understand what it would take or how well we might do. The menu items Cycling Modeling and Scenario Modeling are where these topics are covered.

What data do we need?

Ascent scenario modeling requires just five pieces of data:

  • Ascent – Elevation Gain and Cycling Distance
  • Cyclist – Weight and power output levels
  • Cycle – Weight

Ascent Modeling

Mont Ventoux France (

Ascent modeling computes the effort to make a climb,  independent of the time it takes. Just two numbers would seem too few, but we catch a break because of the nature of gravity and rolling resistance. Aerodynamic drag is usually ignored because ascent speeds are low, and serpentine routes tend to alternate head and tail winds.

The total effort is computed by adding the following two simple calculations:

WorkAgainstGravity = Cycle-Cyclist Weight * Elevation Gain

WorkAgainstResistance = Cycle-Cyclist Weight * Resistance Coefficent * Cycling Distance

This single number provides a complete description of the climb.

Cyclist Modeling

Toolbox: Froome’s Lab Testing September 19, 2017 by Stephen Cheung

Our focus now shifts to the cyclist and their ability to generate sustained pedaling. Cyclists can therefore be modeled by knowing their power outputs over varying time intervals. These numbers  define you and your current fitness level as a cyclist.

They are so important, that Elite cyclists measure them in labs and also using power meters attached to their cycles. While these meters routinely cost several hundreds of dollars, I will show you how you can infer your own numbers from the speed at which you make your local climbs.

Alternatively, you can use the Coggin’s Tables which have used lab measurements to estimate a range of power capabilities for a wide range of cycling levels for both men and woman.

Evaluating Ascent Scenarios

We have a single number defining the work needed to make a climb. but we have not discussed how long it might take. This is where a cyclist’s power levels are critical. If you want to ascend Alpe d’Huze in 37 minutes flat, you will need Elite level power outputs. For the rest of us, we need to lower our objectives until the climbing time corresponds to a power level we can realistically sustain. The practical reality for all cyclists, is their are limits to how long you can sustain high power outputs.

This is where a cyclist’s power output becomes the key to evaluate what it would take to complete an ascent in a given amount of time. If we know what level of power a cyclist can sustain and we know the total work needed to make the climb, we just divide one by the other and we get the length of time that cyclist would need.

We could also take an approach that looks to provide a range of cyclists power levels over a range of time, such as 37 to 120 minutes. This defers the question as to a particular cyclist and instead provides guidelines for a cyclist to know what power level they would need to train to to meet a specific climbing time.

Next Steps

This blog has provided an an overview of how to use scenario modeling to gain practical riding insights. If this is something you would like to learn, you should be encouraged by the simplicity of the process.

The Scenario Modeling section is your key to becoming proficient in  this. The computations are explained and illustrated using several classic riding scenarios. Once you understand this, you will see a way to translate your timed rides into estimates of your own power levels.

Why are Bicycles Rear-Wheel drive?

Are front wheels just “along for the ride”?

Ever notice how your rear wheel wears down more quickly than your front wheel? Or if you put your bike onto a stand and turn the pedals, how the rear wheel turns while the front wheel does absolutely nothing? Does that mean front wheels are only “along for the ride”?

Front wheels play important roles in cycling. Along with the rear wheel, they balance the cyclist weight. They also are  “shock absorbers” encountering bumps ahead of the rest of the bike. And they are a critical component to enabling the cyclist to steer. But when it comes to the cycle drivetrain, it is only “along for the ride.”

Cycle Drivetrains

All cycles have drivetrains including unicycles, bicycles, and tricycles, providing the means for transmitting cyclist pedal power to the designated drive wheel. They come in two flavors. Direct Drivetrains connect directly to the drive wheel, while indirect connects using an  intermediary usually a chain. Drivetrains can connect to the front or the rear wheel, much as cars can have either type of drivetrain. Here is an example of the rear-wheel drivetrain used on modern bicycles.

A modern drivetrain.

What do direct-drive cycles look like?

Direct drive cycles connected to front wheels can be found in unicycles, early bicycles, and tricycles. One turn of the pedals results in one turn of the wheel.

Here is an example. We have a Unicycle doing something real as it is ridden by Cary Gray on his way to South America. The drivetrain directly connects the pedals to the only wheel. (You could  make the argument this is both a front and rear-wheel drive.)

Unicyclist Cary Gray is traveling down to South America on one wheel.

What are the issues with direct-drive cycles?

Drivetrains directly connecting pedals to wheels have important limitations related to cycle speed. Speed is determined by how fast your wheels are turning and how big your wheels are in diameter.

When pedaling and wheels are directly connected, speed is limited to how fast you are can pedal the particular size wheel.  This led to early bicycle designs with large front wheels. It also meant cyclists would fall from much higher heights and led to the expression “taking a header.”

Why are modern cycles rear-wheel drive?

These problems led to the incorporation of gearing which marked a major transformation in bicycle architecture. First of all, it removed the direct connection between pedaling and the wheels, replacing it with a gearing connection. It did so by connecting the pedals to their own wheel called a chain wheel.

Next, the gearing needed to decide which wheel to connect to as the  drive wheel. Bicycles can be designed to connect to either wheel, so the decision to make the rear the drive wheel was certainly driven by wanting to position the cyclist over the front of the drive train.

Gearing and Cycle Speed

Gearing also addressed the direct drive speed issue by eliminating the need for larger wheels. Its solution was to enable the rear wheel to turn in multiples of the pedal turns or cyclist cadence. We will talk more about this later, but here is an example.

Suppose my front chain wheel has 50 teeth and my rear chain wheel has 15 teeth. Then for one turn of my pedal, 50 teeth are pulled per cycle. The rear wheel then must turn 3.33 times as a result. This would then up my speed to 3.33 times than if the pedals and the front wheel were directly connected.

How to use road bike gears efficiently (Pic: George Scott/Factory Media)

What is moving the bike when I am not pedaling?

Bringing in the Experts

This is a question about the fundamental nature of motion. How might we get our answer? Assume we could resurrect two of the great historical figures on the subject, and place them on a curb next to us as we watch a cyclist coast to a stop. How might they explain what we were observing, assuming everyone speaks Greek?

Aristotle’s Answer

Aristotle’s principle of motion was the natural state of the object was to be at rest, and it only moved because something was making it.  Once that impetus was removed, the object would return to its rest state. He would wait until the cyclist stopped and turn to us and say, “I was right!”

Newton’s Answer

Newton would then turn to Aristotle and say “No you’re not.” The cycle slowed down because of rolling resistance and aerodynamic drag. If these were not present, the cycle would have kept on rolling  forever.

Newtonian Inertia

Aristotle and Newton might debate their differing viewpoints as to the natural state of motion of objects, but we know Newton’s held sway because of its ability to explain motions such as planetary orbits that Aristotle’s could not.

Newton said an object’s motion was characterized by a property of all massive objects called inertia, which is a resistance to any change in its current state of motion. An object at rest stays at rest, and an object in motion stays in motion. For either of these to change, forces must be pushing or pulling on it.

Unlike mass which stays the same, the “amount” of inertia an object possesses changes to reflect the current motion. Newton quantified the “amount” as momentum equal to (mass * velocity). The greater the mass and/or the greater the velocity, the greater the inertial momentum.

How our inertial state varies as we ride

During a ride, we are continually changing the cycle’s current inertial state. Newton would observe this was the result of the constant dance between pedaling and the external forces acting on us. When the cyclist stops pedaling, Newton would say our current inertial state will continue to change, but that it is the external forces, not the cyclist, determining the motion.

Why am I able to coast?

We have our answer. What is moving the bike when we are not pedaling, is the bike itself, and its natural tendency to maintain its current inertial  state of motion.  The fact that we cannot coast forever is not the fault of the bike but of the external forces working against it that we are not countering by pedaling.

Triathlete Gwen Jorgensen descending, trailed by her coaches. Credit Garth Milan/Red Bull Content Pool. Gretchen Reynolds AUG. 30, 2016

Coasting is not the only way a cycle moves without pedaling. The other is making descents. Descents differ from coasting in that during the descent, not only do you have the cycle inertia, but you also have a much stronger force, gravity, acting on the cycle.

Both coasting and descending are described using another object property called cycle energy. Pedaling in excess of countering resistive forces stores energy in the cycle via motion or elevation change. This will be a subject of a future blog.  But for now, just note the key role inertial motion plays in making cycling what it is.









How fast do my feet move when pedaling?

Bike Speed and Foot Speed

As cyclists, we always know our current road speed. But have you ever wondered at how fast your feet are pedaling to move you at that speed? It is a simple calculation when you know two things: crankarm length and cadence.

Pedaling Motion and Geometry

Pedaling motion is straightforward. You may be riding for several hours, but in that time, your feet have gone around in circles, pretty much ending where you started. In that time, your feet move around a pedal circle that is determined by your crankarm length.

Cadence is a measurement you can get from your speedometer. It tells you how many complete revolutions you complete in a minute.

Crankarms are manufactured in standard, long, and short lengths. Why the different lengths is another blog. But for a standard roadbike, a crankarm of 172.5 mm or 6.79 inches is in the middle of the ranges. As far as cadences, recreational riders are urged to pedal in the 60 to 80 rpm range, while Elite cyclists will reach 100 rpm and above.

Computing pedal speed

You already have a measure of pedal speed from your current cadence, but let’s use that to get the pedal speed as it moves around the pedal circle.

From simple geometry,  a circle of radius r has a circumference of (2π r). For a crankarm of 6.79 inches, that means your feet travel 42.68 inches or 3.557 ft per pedal revolution. Therefore, the distance covered in a minute is that number multiplied by your cadence. The only thing left is to convert the feet to miles and the time interval to hours to get our foot speed in mph.

Foot Pedal Speed as a Function of Cadence

We can combine all of these numbers into a formula expressing foot speed as a function of  cadence, which produces the graph below where we have plotted if for cadences between 0 and 100 rpm.

FootSpeed(cadence) = (60 * 3.557/5280) *  Cadence  =

FootSpeed(cadence) = 0.0404 *  Cadence

Recreational and Elite Cyclists Foot Speeds

Recreational cyclists have cadences nominally in the 60 – 80 rpm range, while Elite Cyclists are in the range of 80 to 120 rpm. From our chart, that translates into foot speeds of 2.43 – 3.23 mph for recreational rides, and 3.23 – 4.85 mph. for Elite.

Pedal Speed vs. Road Speed

Think about it. Pedaling with a foot speed greater than 4 mph is hard even for elite cyclists. Yet, your cycle is easily moving five times that fast. How a cycle works its magic to accomplish this is something we will cover in another blog.





Can a cyclist reach terminal velocity?

Terminal Velocity Skydiving

If when you are not cycling, you are skydiving, you are already familiar with terminal velocity. It is a condition where the aerodynamic drag balances out your weight and you are no longer increasing speed because the forces acting on you are balancing out.

Saipan Tours Photo

In a nominal skydiving scenario, terminal velocity is about 122 mph. It takes 3 seconds to reach 50% or 61 mph, 8 seconds to reach 90% or 110 mph,  and 15 seconds to reach 99% or 121 mph.

Terminal Velocity Cycling

So what does that have to do with cycling,? I certainly am not doing 122 mph on any of my descents. When I first began cycling, I had a stretch in the Santa Monica mountains where the reward was an extended descent. At first I expected that I would continually pick up speed until I reached the bottom. But what I found was that I seemed to reach a speed around 35 mph where I seemed to stop accelerating.

Joseph Delves 2 May 2017

What I realized was terminal velocity can be achieved even in cycling. The only requirement is aerodynamic drag needs to become large enough to counter the force pulling you down the slope.

When you are on a slope, your weight has two components. One is directed into the road and the other is what is pulling you down the slope. That component is given by Sin Θ * Weight. A 10° slope is  considered steep. For a 150 lb cyclist, the force down the incline is 26 lbs., much smaller than the 150 lbs pulling a skydiver down.

Cycling Terminal Velocity Speeds

At what speed would your aerodynamic drag reach 26 lbs? This happens around 45 mph for a combined cycle speed and headwind.   At half the slope, the required drag would be 13 lbs, and achievable at a combined speed and headwind of only 32 mph.

So the answer is yes!

Aerodynamic drag plays a key role in cycling. It is the limiting factor to speed on the flats, and even more so in descents. When riding down a long descent, your speed will eventually reach a point where it stays constant even though you are continuing to ride down the slope. In another blog, we will discuss what happens when you are riding with a tailwind rather than into a headwind.