I have a question belonging to the picture below. It is mentioned that
whether you walk up or run up stairs the same work is done.
When work equals (force in the moving direction) times the way, then I dont understand why it should be correct. When I run up the stairs I definitely accelerate much faster, while I "gain" more kinetic energy, caused by my velocity. On the other hand when I define my work by the negative difference of potential energy, this statement would be correct. Why is this not a contradiction?
It seems like I am running errors in correctly seperating physical systems, but I can't figure this out. Can you help me out? 
To make my question more precise: Say we reduce the whole thing to a simple straight vertical movement. The Force upwards is given by acceleration times mass. Faster movement upwards must be an increase of acceleration and so an increase in force and that will give an increase of work. Isn't that correct?
Best Answer
Suppose that you are standing stationary outside your house holding a 25kg sack of rice. Now walk 1km to your friend's house to give him the rice. He is not home so you return with the rice. You are now back where you started: stationary again with the sack of rice. Have you done any work? In a day to day sense, you have but you have neither gained not lost kinetic nor potential energy so you have done no work. You may have expended lots of energy but this has all gone to various inefficiencies.
Clarification. What I was trying to address was the distinction between work in its common day to day sense and its sense in physics. In my scenario, you have burned some food to make the trip and conservation of energy will not have been violated. So, the energy has gone somewhere. You and the sack of rice have no net gain in kinetic or potential energy so it has gone elsewhere. Most of it has become heat.
So did you do work? You probably felt that you had since you will be tired after the trip. Your wife who asked you to deliver the rice may think that you have not as the sack of rice is where it started. In physics, work has been done and the conservation of energy has not been violated, it is just that you might not notice or care where the work was done.
Back to your example, you might or might not have done more work by running up the stairs. You would need to determine the energy that went elsewhere than your kinetic or potential energy. This is possibly more a question of biology than physics. In which scenario is your body more efficient?