Bicycling Uphill

Not long ago I was huffing and puffing on my bicycle up a very steep hill and another cyclist blew by me with apparently little effort.  Now, in addition to looking younger, stronger, and having a better bicycle, he appeared to be carrying significantly less excess, ahem, baggage.  So I got to wondering how much difference one’s weight (or to be charitable, a loaded backpack) makes when ascending a hill on a bicycle.

The power required to move a bicycle at constant speed can be grouped into three parts.  There is power required to balance rolling resistance, to balance air resistance, and to increase the elevation.
 
Rolling resistance is a pretty complicated phenomenon that depends on the type of road surface, the type and shape of tire, the inflation of the tire, and the speed of the tire relative to the road.  Compared to air resistance, the dependence of rolling resistance on speed is relatively weak.  Partly because of this, but mostly to make life easier on ourselves, we’ll neglect that dependence, and, for our purposes, take rolling resistance as a constant.  Browsing around the internet, it appears that a reasonable range for typical bicycle rolling resistance might be about 12-30 W.  There are almost certainly situations that fall outside of this range, but it is probably representative for most cases.

Air resistance can be described by this equation:

where F_d is the drag force from the air, ρ is the air density, C_d is the drag coefficient, A is the total projected frontal area of the bicycle and cyclist, and V is the velocity.  In order to get the power required to maintain a given velocity, we have to multiply the drag force by the velocity which leads to this expression:

For something as complicated as a bicycle and rider, both the drag coefficient and the projected frontal area typically must be measured.  Again, browsing around the internet, it appears that a rough representative range for C_d*A might be around 0.4 to 1.0 m².

The power required to increase the elevation at constant velocity is:

where m is the mass, g is the acceleration of gravity, V is the velocity, and grd is the grade of the road.  While this expression applies to any climb, it can be simplified considerably for the case of bicycling.  For any road grade less than 14% the error in replacing “sin(atan(grd))” with just “grd” is less than 1%.  Considering the large approximations that we are making with rolling resistance and drag coefficient, introducing an additional error of less than 1% shouldn’t hurt too much.

Now, we can write the total power requirement as:

Examining the three terms, we are taking the power to overcome rolling resistance as a constant that doesn’t depend on anything.  The power to overcome air drag depends heavily on the velocity, but not at all on the weight (i.e., the mass). The power to climb depends linearly on the velocity, and linearly on the weight, and approximately linearly on the grade of the climb.  Using C_d*A=0.7 m², and rolling resistance as 21W, the total power looks like this for two different velocities and three different grades:

It is pretty clear that the power requirement goes up fast with increasing weight at a 5% grade.  This figure shows the breakdown for the 3% grade and 12mph line in the previous figure.  

Neither the power required to overcome air resistance nor the power to overcome rolling resistance depend on weight, but the power required for climbing depends strongly on it.  This is good news for those with a heavily loaded, um, bicycle, as long as they stay on flat ground.