Recently, I've been thinking of ways to maximize my average speed during road rides. I had previously heard that it's most efficient to have a steady power output up and down hills, but since one spends more time going uphill than downhill, it seems that to go faster, one should expend more energy on the uphills. My question was how much more energy?
To find the answer, I set up an equation to calculate total energy expenditure over a symmetrical hill, of slope alpha, including the work done against and by gravity along with wind resistance, using the average speeds uphill and downhill.
E = [work uphill] + [work downhill]
E = [m*g*sin(alpha)*d + 0.5*C*p*A*d*v1^2] + [m*g*sin(-alpha)*d + 0.5*C*p*A*d*v2^2]
where:
E = energy
m = mass
g = gravity
alpha = slope of hill
d = length of one side of the hill (hypotenuse)
C = drag coefficient
p = density of air
A = frontal area as related to drag coefficient
v1 = average uphill velocity
v2 = average downhill velocity
I condensed the constants/knowns and split up the velocities into distance/time to come with the generic differential equation below.
C1 = C2[1/x^2 + 1/y^2]
where:
C1 = Energy
C2 = 0.5*C*p*A*d^3
x = time uphill
y = time downhill
Solving for y;
y = [C2*x^2 / (C1*x^2 - C2)]^1/2
I then plugged this into Matlab using somewhat reasonable values for energy and distance and a range of x-values from 210 to 1400. I plotted x vs y, or time uphill vs time downhill (figure 1). I added the arrays to find average velocities, then found the maximum value in the new array of averages, and finally the ratio of time uphill to time downhill for the maximum average velocity.
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| Figure 1: Time uphill vs time downhill |
Even though the answer isn't revolutionary, there are a couple conclusions to be drawn from this exercise.
If you're trying to go as fast a possible, put significantly more effort into the uphills. The problem with this is that it is highly inefficient and not sustainable for long. It's much more efficient to have a steady power output. To find this optimum, breaches the realm of biomechanics, and I would guess, would be very difficult to truly quantify and would involve extensive testing for each individual. Knowing the terrain and your own limits and how quickly you can recover will get you a long way though.
Another conclusion is that it's more worthwhile to improve your weaknesses than to improve your strengths: spend less time going slow. In mountain biking, bike selection can be critical in this regard. For example, I'm great at riding uphill, hardtail or full suspension doesn't make much difference. But I lose a lot of time on the downhills, so full suspension it is. I also get tired accelerating out of corners, so I went with 27.5 inch wheels rather than 29. Also, full suspension 29ers are a lot of bike for someone my size and can be difficult to maneuver quickly; however, there are situations where the stability of a 29er would be advantageous (like the fast, flowy trails in Colorado).
There are other fun things to be done with this equation such as finding the ratio of one's power output that goes into overcoming wind resistance compared to riding up hills of various slopes. But I'll leave that for another time.
