I posted a similar question yesterday, but the question hasn't been solved yet, so I'm posting a similar question again. (sorry for similar question..)
Let me write my interpretation of the situation below.
First case) On frictionless ice, 50kg of people exert force on 200kg of objects. (Initial speed is zero.)
If the later speed of an object weighing 200 kg is 1 m/s, a person weighing 50 kg will have a speed of 4 m/s in the opposite direction.
The kinetic energy of an object is 100 J and the kinetic energy of a person is 400 J, and the total kinetic energy is 500 J. This means that a person has consumed 500 J of energy.
This is true even if a person does 100J of work for an object, and an object does 400J of work for a person.
Second case) It's a similar situation, but let's think about a slightly different situation. A 200-kilogram object is on frictionless ice. Let's say that someone pushes this object by 1 meter with a force of 100 N.
And what a human does to an object is 100 J as above. The later speed of the object will be 1 m/s as above.
The object will also push a person by 1 meter with a force of -100 N (reaction force), so what the object has done to a person is -100 J.
Something's wrong.
What a person does to an object is equal to 100J in the first case and in the second case. But in the first case, the total kinetic energy is 500 J, which is about as much as 400 J.
Where did the error come from?
Best Answer
You are just wrong. Sloppy in the analysis of the issues. And if you have already asked the question before, do not make new ones.
In the first case, the person converted 500J of chemical potential energy (stored food) from muscles and gave 100J of that as kinetic energy for the object, 400J leftover is the kinetic energy of the person. Conservation of energy is perfectly fine with this.
In the second case, the person converted only 100J of chemical potential energy from muscles and gave all 100J of that as kinetic energy for the object. Again, conservation of energy is obeyed.
If you have ever played in an ice skating rink you will know that it is far more difficult to push in the first case than in the second case. There is no paradox at all. The forces, impulses, distances, energies, will have a sensible resolution.