The basic question is, is there an amount of slope (rise/run) where it is less efficient to try to bicycle than it is to walk or run?
My intuitive feel from personal experience is that a bicycle is vastly more efficient than running on flat ground, but as one goes up steeper slopes, the advantage of the bike disappears and the extra weight of the bike is a hindrance, and at some point of steepness, it is better to leave the bike behind.
My guess is also that this should be a calculable value, especially if the parameters of the bike weight, gearing, rider/walker weight, and power level are known.
In my first post I supposed that I could get some reference by researching performance of top athletes. This gave me a graph and a starting point. I have added some more data points in the graph below:
The problem with this graph (besides having admittedly too few data points) is that when I research the common physics functions that are used to calculate such data, the formulas are not curves, they imply lines, and the effort differences would not develop intersecting graphs.
The basic energy equation is of the form E = mgh (potential energy = mass * gravity constant * change of height).
In my previous post I supposed that the effort of balancing a bike becomes a nonlinear effort as the grade becomes steep. Certainly if the gear ratio is fixed and the energy expenditure has some natural limit, there will be a slope that a bicycle rider cannot climb. But in physics, bicycle riding at a slow speed is a form of work that is based on a principle of conservation that says that the total work is independent of the path. Essentially, a frictionless ball rolling up and down hills has a net energy change that only depends on the beginning and ending point. Even when I introduced friction and balancing gymnastics, the functions do not create an intersecting set of curves as the above example seems to imply.
The initial idea I posited was not getting support from the physics that I was using. Then I found this post:
09172012, 10:13 PM
 
 
The answer is that a bicycle takes advantage of both muscular and mechanical efficiency. There is a very good article that explains this (Tucker, V.A. The Energetic Cost of Moving About. American Scientist, JulyAugust 1975.) I can't seem to find this online, but I do have a copy that I just scanned. If anyone wants it, just send a PM with your email address and I'll be happy to send a copy. The very first paragraph of this paper reads much like the OP, and the explanation is quite interesting.

The article is a very good read. Boiled down to my paraphrasing, Mr Tucker shows that walking on flat ground is not very efficient at converting muscle energy into forward horizontal motion because the force that a standing person can generate depends on the vertical pull of gravity, but the forward resistance to motion is at right angles to that. A bicycle is better at converting muscle energy into a pushing force because the pedal efficiently opposes the force of gravity and converts that force into forward motion through the crank and chain and friction of the tire. In terms of efficiency, walking converts 24% of the consumed energy into forward motion on flat ground, whereas a bicycle can convert 2125% of the consumed energy into forward motion on flat ground.
To get a basic sense of the force transfer problem, sit on a floor with your back next to a wall and feet on a smooth floor. Push hard against the wall. Now sit in a narrow hallway where your feet are able to push against the opposite wall. You will generate much more force against the wall in the second case. In the first case the floor is tangential to the resistance of the wall. In the second case the two walls give opposed surfaces and help the muscles create much more force.
From this inspiration, I realized that as one climbs a slope, the resisting force of motion (up the hill) is vertical compared to gravity and the walking person's muscle efficiency can start to replicate the efficiency of the bike pedal as the slope gets steeper. When one climbs stairs, the resisting force of the step is perfectly in line with the generating force of gravity.
I created a simple graph for a series of increasing slopes. For the bicycle energy I used a standard spread sheet based on Bicycling Science: David Gordon Wilson. For walking I split the energy of motion into two components: One is the energy needed for walking on flat ground (which uses a fairly constant amount of energy based on the weight of the person). For the second, vertical, component of the walking energy demand I used the standard potential energy equation noted above, E = mgh, and I presumed the same energy efficiency as is used in the case of the bicycle. I used a constant slow speed. As the slope gets steeper, the proportional energy conversion efficiency of the walking person increases, and eventually the lower weight of the person walking as compared with the person who is also moving a bike starts to give the advantage of energy requirement to the walker.
In the graph the blue line for walking is the sum of the horizontal and vertical components of the energy demand.
From this graph I claim that as a slope goes steeper than 10 to 15 percent, the bike is no longer an advantage, either for speed or for efficiency. You will note that the 1015% estimate is somewhat lower but close to the 1819% intercept predicted by the first graph.
As a further cooberation of this estimate, in Bicycling Science by David Gordon Wilson, in pages 164166 ("Power and Speed" Chapter) the author asks "Should one walk or pedal up hills?" In that section the author also investigates the efficiency of energy conversion by the muscles. The book states: "Calculations show that the 15percent gradient may be a critical one, and that at gradients of 20 percent there is no appreciable advantage to riding the bicycle, even in low gear."
With this mathematical conjecture in hand, I did a small field experiment. I found a gravel slope of 16%, jogged the slope, bicycled the slope, and walked the slope. In each case I timed my effort. I could not really control for a single speed or for a similar power output, but with two repeated efforts I found that my time for jogging compared with my time for bicycling actually showed a somewhat wider advantage for leaving the bike behind than predicted above. The fact that the available slope was rough gravel instead of smooth pavement probably further decreased the advantage of the bike. The bike I used did not have extremely low gears, so the test needs further development.
The "Climbing Rate" graph above duplicates the graph in my previous post with added data from Pikes Peak Marathon (7700 ft elevation change in 2 hrs and 1 minute, 15% slope), Bear's Reach Speed Climbing (400 feet elevation change in 4 minutes and 25 seconds, 400% slope), the Tour de France Col du Galiber (1100 meters elevation change, 16.7km at 20km/hr, 15% slope), and Lance Armstrong riding Waipio hill in Hawaii (800 ft elelvation change in 9 and a half minutes, 25% slope).
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