Big OOPS on my part that obviously causes a lot of confusion. That should have been written "I've got NO problem...".
Completely different meaning there!
True, but is completely irrelevant to anyone who is not trying to conduct the most precise of experiments. As far as everyday activities are concerned, you're pretty much good with the 10m/s/s approximation.
That continues to be contradictory! Gravity is NOT one of the vectors that is combined to give forward motion. Draw a force diagram!
Forward motion is entirely due to the horizontal force applied through your foot that is countered by the frictional force between your foot and the ground. That is why you wear spikes on the track and can't move much at all when you are wearing dress shoes on ice. Gravity is the same in both cases.
After you leaned forward and fell, is your center of mass higher, lower or the same as before?
Anser: lower. How do you get it back to the starting height? If you're Jack on the POSE forums, apparently you can then fall back up. The rest of us live in a world where you must push through your legs to drive your mass at an upwards angle that therefore has a horizontal (forward motion driver) and a vertical (paying back gravity with interest) component.
Running you can either fall and then spend energy to raise yourself, or, as it appears all elites do, raise yourself (while primarily driving forward) and then 'fall' back to the next landing.
Chart the center of mass of any elite runner and you will see that their center of mass is highest when they are completely airborne in the middle of their stride.
This is in complete contradiction to your POSE description of 'falling forward'.
If you are not providing a horizontal vector when running, then any forward lean or push will result in your feet flying backward while your center of mass has no net horizontal motion. End result would be you flat on your face on the ground exactly where you started from.
Since that is obviously not happening, there MUST be a force in the horizontal direction imparted through the point of contact with the ground. That would be the frictional force between your shoes and the ground. It also explains why ice skaters need to angle their feet when they want to move forward since trying to imitate walking motions would provide minimal horizontal force due to the noticeable lack of friction.
Done! - clarified my mistype from before and introduced you to the concept of friction.
My slip-up and missing that crucial 'no' has caused you mounds of confusion and for that, my apologies.
For clarification purposes, I'll defer to JimFiore's question asking for a clear and concise explanation of whether POSE specifies that "gravity does the work" or not.