Ever since learning to integrate in high school Calculus, I've been curious as the the physical meaning of the integral of position w/ respect to time. Over the years, the topic has come up again and again in my mind, and have finally decided to put an end to my pondering; or at least let others ponder with me.

 

Is there any firm grasp of the concept of the meter-second already in existence? Is there a good use for this?

 

I have come to call the integral of position wrt time "longevity" as that seems to be the best term in my mind to describe it. From what I have been able to determine with some simple thinking is that longevity can be related to amount of energy or force placed on object in a gravitational field (ie. if you put a high amount of kinetic energy on an object, it will have a larger longevity than if you put a small amount of kinetic energy on an object).

 

Any thoughts on this?

Where have you ever seen position integrated with respect to time? Not that you can't do it, but there's no physical reason to do it.

That is precisely my interest. People don't do it. You aren't taught it in any high school or college classes. My question is, has ANYone done it and then tried to find a physical application for it? Surely somewhere, there can be something that makes use of it.

 

In the present scientific world where space and time are intertwined within one another, I figure there must be some sort of valid use of the integral of position wrt time. Just wanting to get folks thinking and see what ideas people can come up with. If in the end it turns out to be useless, so be it, but it is quite fun to contemplate.

My apologies, this is in the wrong discussion board... moved to physics:

http://www.scienceforums.net/forums/showthread.php?t=18441

I don't think that anyone has tried to do anything with it. The reason would be that state variables that appear in the laws of physics arise from the analysis of a physical system, as opposed to being forced into them. The dynamics of a mechanical system requires that [imath]x[/imath] and [imath]\dot{x}[/imath] be used as state variables, because potential energy depends on the former and kinetic energy depends on the latter. The reason [itex]\int xdt[/itex] isn't used is because there is no form of energy that depends on it.

 

Of course, there is nothing stopping you from re-writing [imath]x[/imath] as [imath]\frac{d}{dt}\int xdt[imath], but why would you want to do that?