The Physics of Cycling, or Why I Suck on the Uphill

For all you non-nerds out there - avert your eyes!  This is the nerdiest post you will ever find on a cycling blog.  The bulk of this post is about the physics of cycling paying special attention to hill climbing.  Bored all ready?  Go read something else - maybe this.

Still with me?  Cool.  I am reading a wonderful book written by David Wilson entitled "Bicycling Science, 3ed".  He must be an OCD physicist who is really into cycling.  This book has everything related to cycling and more including but not limited such interesting/obsessive topics as: fast twitch vs slow twitch muscle fibers, properties of commonly used bike materials, gearing, heating of rims during braking, and the most efficient way that humans move (hint, its the bicycle).

I was particularly interested in the physics of rider/bike weight related to hill climbing ability.  Here is the master equation for cycling which relates rider power to bike velocity with effects such as rider aerodynamics, rolling resistance, wind speed, incline, bike and rider weight included:

where

W = Power delivered by the wheel (slightly less than what the rider produces due to losses and coupling efficiencies)
K_a = drag factor (kg/m) 0.1 to 0.3 small recumbent rider 0.1 large upright  rider 0.3, typical values 0.2-0.25
V = bike velocity (m/s)
V_w = wind velocity (m/s)
m_r = rider mass (kg)
m_b = bike mass (kg)
S = slope in (%grade) so 6% grade = 0.06
C_r = rolling resistance 0.02 (racing tires) - 0.08 (MTB tires), 0.003 typical for road bikes
g = gravational constanct = 9.81 m/s2

Before we proceed, it is helpful to discuss what rider power output actually means.  I have never trained using power, so all these values come from the book.  Supposedly a recreational cyclist on the flats usually rides at about 100 W.  A fit cyclist can maintain 200 W for hours.  The power output of Marco Pantani (an excellent Italian hill climber) who climbed the Alpe d'Huez stage of the Tour de France that has an average slope of 0.079 (7.9%) for 13.84 km in 38 minutes, sustained an output of 400W that whole time.  So I am guessing that realistic values for power output to be 200W to maybe 300W for the average weekend warrior.

Playing around with this equation in Matlab lets us play with a lot of different scenarios.  For example, what happens if we swap out road tires for mountain bike tires keeping all other things even.  The result?  For the same power output by the rider, you speed increases by ~3 mph going to the road tires.  [Most of the calculations below are for my weight and best guesses for other parameters, obviously yours would be slightly different, but the trends would hold.]

How about the effects of headwind?  Suppose there is a 10 mph headwind (not uncommon here in NM). Riding on the flat at 15 mph takes about 100W, while maintaining 15 mph in 10 mph headwind now takes 240 W.  Conclusion, wind bad.

What about the effect of aerodynamic drag?  A small rider in a recumbent bike has a K_a of 0.1, while a large rider on an upright bike has a K_a of 0.3.  Most road riders have drag factors of 0.2-0.25, where 0.2 might be for a tucked rider where the small of his back is the highest thing on his body.  So what does going from upright position to tucked position gain you?  For a 100W output, you would jump from 14.3 mph to 16 mph.  Not bad.

Now we get to the meat of my analysis - weight and hills.  My steel frame beauty, Guiseppe, is not light by any stretch of the imagination.  It is a huge frame and has heaver than normal wheels.  What would happen if I found a magic bike that was half the weight of my bike?  On the flats, nothing really.

How about on a hill - a fairly steep 10% grade hill?  If I rode a really light bike what would happen?  Turns out, not much either (plot below).  If my climbing power output is 300W, then halving bike weight would increase my speed from 5.25 mph to 5.66 mph.  If my power output is 400W (which it isn't), then halving bike weight would increase my speed from 7.00 mph to 7.5 mph.  So minor improvements.

Now lets look at the effect of rider weight (plot below).  Suppose we had two riders, one weights in at 235 lbs (me) the other at 140 lbs (not me).  If both riders are pedaling up a 10% incline at 6 mph, the heaver rider is expending 334 W compared to the 216 W for the light rider.  It is hard to be a big fat man. 

With a little manipulation you can turn the above equation around so that you can fix power and see how your velocity varies.  This is done for a heavy (235 lbs) versus a light (130 lbs) rider in the plot below. If you assume that most recreational riders are about the same power output level (which may be a good guess), the difference in speed on the hills solely comes down to weight.  For example, supposing two riders can output a sustained 300W for a hill climb.  The light rider will go 8.1 mph while the heavy rider will go 5.2 mph.  In order for the heavy rider to keep the same pace as the light rider, he would have to have a power output of >400W - or in other words be a stronger rider than Marco Pantani mentioned above.

That's not gonna happen.  So enjoy it you light weight skinny people!  Physics is on your side on the hills!

Enjoy this?  Check out the companion post about why going down hill is awesome for big people!