how much power do you need to go fast?

There is a compelling reason why the old 55 mph speed limit made sense if conservation was a goal.  The power required to move something through a fluid (air for example) increases as the cube of the velocity.  At slow speeds the power required to move may be dominated by other effects and, in cars, there are several complicating factors that need to be considered, but once you are past a certain velocity, air resistance is the major force to overcome.

It sometimes surprises people to learn the power requirement is proportional to the cube of velocity - in fact if you ask a lot of college educated people (I have) they'll guess it should go as the square.  They are probably confusing energy with power.  Power is simply the rate at which energy is "used".  So if you have a gallon of gasoline you can get a lot of power by burning it quickly in your engine with the throttle wide open or you can produce lower levels of power and consume it at a miserly rate.

Here is a simple physical explanation of why power goes as the cube.  Consider something moving through the air that presents an area to the air.  You can approximate the amount of energy being transferred to the air in some time by calculating the mass of the air moved aside in that time.  Now calculate the kinetic energy (rho is the density of the air, so the mass is just the volume of the shape traced out in time t, and the velocity is the velocity the object is traveling relative to the air) and divide by t to get the power.  

Simple and sweet...

It turns out bikes at normal cruising speeds - say about 10 to 14 mph - are in a region where most people can develop enough power to move easily, but increasing the speed to 20 mph requires serious effort.  Changing the effective surface area of the bike is a way to deal with the challenge, which is why you bend down into a more aerodynamic shape.  Most cars begin to see serious power requirements for moving air aside at around 40 or 45 mph. 

more on this topic later - we'll talk about how you can get the equivalent of 1000 miles per gallon on a bike and even more on an e-bike. 

 

(I'm just talking about a cross section of air here.  If you move a body through the air its shape is important and described by C_d - a coefficient of drag.  The area in the equation becomes C_d * cross section area of the object)