How to calculate the
speed of a commuting bicyclist
Equations
Cyclist reach a steady speed when the motive
power that they produce balances the rolling and air friction; in other words
when the friction drags balance the motive
power.
The air drag is given by the well known
equation:
Here A is the frontal cross section area, Cd is the drag
coefficient, ris the density of air, The air drag is proportional to the
velocity v squared. The dependence on the cross sectional area should make
intuitive sense; its harder to shove a big object through the air than a thin
object. The density comes into the air friction because the moving object has to
shove the air out of the way.
The rolling drag is given by the
equation:
Here M is the mass of the rider and bike, g
is the acceleration of gravity, and Crr is the coefficient of rolling friction.
This is simply the weight (mg) of the rider and bike times the coefficient of
friction.
The total drag is the sum of these two
drags:
The power required to overcome these drags
is the velocity times the drag:
Thus the power required to overcome the drag
scales as the velocity cubed.
To calculate some actual values, we need to
know the values of constants. The density of air and the acceleration of gravity
are well known.
The rolling coefficient is more obscure, but
various sources give it as approximately
Certainly one can due better than this with
a tuned racing bicycle, but the average commuter bike isn't in great shape.
Fortunately the exact value doesn't matter much because the air drag is much
more important than the rolling drag at high
velocities.
The drag coefficient is commonly take to be
0.9
The cross sectional area is more
problematic. Most of the measurements have been taken for racing cyclists, not
commuters. Typical values quoted are 0.4 to 0.6m^2. Since a commuting cyclist
rarely crouches, and is probably more upright than a racer using the top of her
handlebars, I will use 0.67m^2.
Finally, we need the mass of the cyclist and
bicycle. Take a 150lb cyclist, with a heavyish bike of
28lbs.
Velocity vs. Power
curves
Using these equations, we can easily
calculate the speed of a cyclist as a function of the power the cyclist is
putting out. For example, if the cyclist puts out
100w
Here are two graphs of the power that a
cyclist has to put out as a function of the cyclist's
velocity.
Thus much over 20mph (10m/s) is very
hard.
There are many calculators on the web that
will tell you the power for any given velocity. Be aware that they do not always
state the constants that they are employing. Two such calculators are
at:
A very good shareware program to calculate
the power, (and many other things) is at
Stop signs and
bicyclists
Frequent stop signs dramatically decrease
the average speed of bicyclists. For instance, it turns out that the to maintain
as speed of 12.7mph, the cyclist would have to increase her power output from
100 to 500watts on a street that has a stop sign every 300ft. Calculating this
exactly is messy, so heres a simplified version:
Defining the distance between stop
signs
The time between stop signs
is
So every 16.1 seconds she would have
generate, and then lose, all her kinetic energy. Now since she slows down near
every stop sign, her peak speed must be higher than her average speed. Lets say
she can accelerate and deaccelerate at a maximum rate of 0.15g. For comparison,
a car going from zero to sixty in 13 seconds has an acceleration of
0.2g
Acceleration
The approximate peak velocity can then be
found by solving the equation:
The solution is
Her kinetic energy at this speed would
be
She has to recreate this energy every 16
seconds, so her average power for this alone must be
about
At this speed, the power necessary to
compensate for the drag would be
Thus the total power she would have to
supply would be
Note that this is very approximate. In
particular, she is not at her peak speed all the time, do the drag power would
be lower. Nonetheless, approximately the same answer is obtained when the
calculation is done more exactly.
