Okay, so looking at the basic definition of energy the force is summed over the distance it is applied. Why exactly is it taken over the distance applied and not the time applied? I understand that the impulse and hence the change in momentum is what we call this summation over time, but it's not exactly clear to me why we chose to do it the way we did.
Side note - I've seen the examples where there's an object at rest and if you took the force over time you would get infinite energy, but if you took the sum of the force (both holding it up and pushing it down (gravity) you would get 0, and thus an integral over the net force would be 0, right? Thanks
Okay, so really the crux of what I'm getting at that I can't seem to find an answer for is: Why did we choose to do it this way? What experiment or thought experiment led us to believe that momentum isn't in fact energy, but a separate quantity? The problem I keep having is that when you want to sum up this quantity we know as force, you have two options, sum it over the time its applied or the distance. I just don't understand the idea behind choosing one over the other.