The Physics of Cycling II, or Why I am Awesome on the Downhill

[As a few people correctly pointed out, my previous analysis is for constant velocity - these totally neglect acceleration.  When you consider acceleration, more things come into play such as rotational inertia of the wheels, etc..  That is a whole other problem entirely.  Maybe next time...]

So believe it or not, while pedaling my butt around northern New Mexico this weekend I thought about the physics of cycling.   My last post talked about the big disadvantage heavy riders have on the uphill. But for every uphill there should be a downhill right? So this post will talk about the advantages of going downhill when your a big fat guy like myself.

So we will use the previous equation that talks that takes into account such things as rolling resistance of the tires, drag factor due to aerodynamics, the effects of headwind, and slope of the road. That equation still holds, the only difference is that our slope is now negative.

When we plot up some downhill slopes for two different rider weights, we see some interesting trends. As shown in the plot below, you will notice that the power output dips negative before increasing as the speed increases. This means it does not require any work by the rider to achieve speed on a down hill. In the case below, up to about ~12 mph both riders can just coast along using the slope. If the rider just coasts and doesn't pedal (when power output equals zero), this will be his terminal velocity on the slope. In the example below, the heavy 235 lbs riders terminal velocity is 13 mph compared to 10.5 mph for the 140 lbs rider. Supposing they are both cruising along expending 200 W (the output for a relatively fit rider), the heavy rider will go at 23.5 mph, the light rider at 22.5 mph.

 Things start to diverge as the hill gets steeper. For a 2% downslope, the terminal velocities are 16 and 20 mph for the light and heavy rider, respectively. If both expend 200 W, the velocities are 25 and 27 mph for the light and heavy riders, respectively. To think about it another way, if both riders are cruising beside each other at 25 mph, the light rider is expending 200 W while the heavy rider is only expending 130 W.

Things really start to diverge as the slope gets steeper. For a 4% downslope, the terminal velocities are 24 and 30 mph for the light and heavy rider, respectively. If the light rider wants to match the heavy riders terminal velocity speed of 30 mph (which requires no work for the fat guy), the light rider has to pedal at 200 W.  This my friends, is the origin of the phenomena known as "getting Scrymge'd".

Now lets just look a just terminal velocity. Assuming that wind speed is zero, the equation from the previous post can be rewritten as

where

V_t = terminal velocity in (m/s)
K_a = drag factor (kg/m) typical values 0.2-0.25
m_r = rider mass (kg)
m_b = bike mass (kg)
S = downslope in (%grade) so 4% grade = 0.04
C_r = rolling resistance, 0.003 typical for road bikes
g = gravational constanct = 9.81 m/s2

which is the equation for terminal velocity of a cyclist.

The terminal velocity of two cyclists who again weigh 140 lbs and 235 lbs is plotted in the figure below as a function of down slope. For small slopes (<0.5%) they go about the same speed but then rapidly diverge. As one can see, the terminal velocity for the 235 rider is much faster than the lighter rider!  For a sanity check, my top speed on the descent from Truches last weekend which had long sections of 10% grade was about 49 mph. I didn't pedal, nor did I tuck that much (which would have reduced my drag or K_a). By the plot for a 10% grade my speed should have been about 47 mph. As we would say in science - pretty good agreement!

What happens if I would tuck compare to cruising down in a comfortable riding position?  In the comfortable riding position, my K_a is approximately 0.25.  If I could myself more aerodynamic so that my back would be the highest point of my body would make my K_a approximately 0.2 (which is the K_a of a professional road biker in a tuck which I probably couldn't achieve).  What happens?  I could squeeze a few more mph out of the descent - my Truces descent of about 10% would go from 47 to about 52 mph.  So tucking actually does make a difference.  To make your K_a even lower requires body suits and funny helmets or even teardrop shaped recumbent bikes.

If you are curious about your own terminal velocity check out the 2D plot below. The x axis is rider weight and the y axis is the slope. The color on the plot corresponds to your speed. So say you weigh 180 lbs, to figure out your speed put your finger on 180 lbs and move your finger straight up to see how fast you are going on a given slope. If I head down a long 15% slope, my 235 lbs makes my terminal velocity almost 60 mph.  Yikes!

Google tells me that supposedly the one of the typical fastest down hill speeds is ~74. (rider JJ Haedo did 72.7 mph on the Giro de Italia in 2006, and the fasted Tour de France speed is 74.5 mph - but I can't find any reputable sources on that). Apparently, the pros get a little scared at speeds north of 50 mph so most descents are about that average.

Contact me for the Matlab programs that were used to were generate these plots.