# 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 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 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.

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  =

### 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.