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I am a little stucked in understanding the connection between kinetic energy and work $W=\Delta E_{kin}$.

This is a second to second thing so:

Let's say someone is pulling a box. The box is at rest when he starts and comes to rest when he stops to pull. If I would then investigate this process I would come to the conclusion that it has no kinetic energy (v at the beginning and at the end ist equal 0). And as you can see - it is at rest so the statement "is there".

Question (1): Does this count as work done ? Since $W=\Delta E_{kin}$ there is no work done, isn't it ?

But (!) If you cut out this little tiny interval where the box stops and rests and only look on the interval in which it actually moves:

While we move the box we are doing work on it and while it moves it has a certain amount of kinetic energy.

Is that correct?

Sorry if this may sound rediculously easy but I am yet a little confused

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Not at all, this is a very important question. Not for physics, but for you: it is very important to understand these things.

You almost got it, but there are a few ideas you need to take into account. The most fundamental is

You must think of "work done" as "work suffered by the object", regardless of what you as a person do on it.

  1. According to physics laws, if you push an object, it will keep moving indefinitely forever (in straight line and constant velocity)
  2. If you push an object, but it stops after you stop pushing, then there must be another force you're not taking into account. (Friction!)
  3. So this is what is happening. On a frictionless ideal surface, you would push an object and it would acquire a kinetic energy given by $E_k=\Delta W$, so it would get the speed $v=\sqrt{\frac{2W}{m}}$.
  4. However, friction is acting to stop the body. This means that the work done by friction is the same as the one you did, so they cancel out and you have no movement. No kinetic energy means no work.
  5. But "no work" means "no NET work ON THE BODY" under consideration; i.e. the body received the same amount of work both negative and possitive. But that does not mean you didn't get tired. You usually provide the possitive part, and nature fights it adding the negative part. That's why you won't make work pushing a wall, but you'll get tired.
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