I have chosen this example as a paradoxical limit, but my question really regards the optimization, distribution and outcome of force when you try to pull a weight: how can you optimize your effort/energies spent?
In order to pull a weight you need a rope and a firm grip on the ground. Suppose this last problem is solved 100% (like in the second picture) preventing the feet from slipping. The rope is perfectly horizontal, suppose the man's weight is 100kg. The angle may be between 0° and 45, at your choice. Now, you can decide you want to keep:
your leg(s) firm and stretched and apply your force to the rope pulling it like in this picture.
Suppose you are exerting a force (on the train) of 300 N, how much force is going to the train and how much is going to the ground ? and, how much energy is being wasted? Is it possible to improve the percentage of energy which is actually transmitted to the train? (without tricks or external factors)
You can stand (like in this picture but holding a rope in both hands) with bent knees and keep your arms stretched. Then you stretch your legs and apply a push on the ground, in this case you are exerting 300N directly on the ground and indirectly on the train , as the displacement of your trunk will pull the train.
Of course this position is most favourable a): because the legs are usually stronger then the arms [probably, force is more than double here] b): because the angle is near 0°. That's why this man can pull a train!:
but all other factors being equal, how does push-on-the ground affect the distribution of forces as compared to pull-on-the rope?
Which technique is more effective/favourable, which makes the train move with less energy spent? Is the outcome exactly the same as in the previous case? if not, why? what physical principle makes the difference? Can you draw two sketches and show how forces interact and are distributed?
Supposing you are exerting a force... how much of it is going to the train and how much to the ground?
This is difficult to answer since you seem to have a major misconception about forces. The best answer is to go back your freshman, or even high school, physics book and read the section on forces...Olin Lathrop
I am doing my homework, but this site is all about clarifying, explaining misconceptions. I regularly help students (like: here to dispel theirs and, lo their misconceptions do not seem major to me, nor trivial, everyone got his own, sometime or other!.