Problems with scale

[putongren]

I'm not sure where to put this post, but it has related to relativity somewhat, so I posted it here. The question I have has more to do with philosophy of science. Anyway, according to Newton's, p = mv, F =dp/dt = ma. But according to Special Relativity, p = γmv , F = dp/dt = γ3maparallel + γmaperpendicular.
The derivation of force in terms of special relativity can be seen in the special relativity wiki section.


So if I were to analyze force at very high speeds, I would use the F = γ3maparallel + γmaperpendicular since that takes relativistic effects into account.

So my question is that are scientific equations valid for only certain scales? Newton's force equation should be treated as only approximation and practical for problems at low speeds. It seems to me that the other force equation is ideal for high speeds. So do you think that the progress of science (physics in particular) is just trying to come up with equations for particular experiments, and modify it until new experiments cast a new light on the theoretical side?

[CompuChip]

In a sense, yes. The following image shows it rather neatly IMO (it's everywhere on the 'net so I suppose I'm allowed to post it here as well):
http://scienceblogs.com/pharyngula/upload/2007/02/science_flowchart.gif

By now we understand that, whatever theory we come up with, it most likely has a range of validity. So we're always searching for theories with different applicability ranges. Of course, if such an applicability range overlaps with that of an earlier theory, which has proven itself to work very well, we would like to see some (mathematical) limit such that the new theory reduces to the old one on the overlap range. Preferably, you don't get exactly the same, but something very slightly different -- some effect that can only be seen in a very accurate measurement, for example -- that allows you to test the new theory.

There is always an interplay between theory and experiment. On one hand experimentalists come up with new results which theorists try to explain by "modifying equations" in a way that makes some kind of sense (e.g. we can of course just define an equation that gives the right answer, but that's not what we usually mean by "doing physics"). On the other hand theorists come up with new principles as to how nature works, and experimentalists test this by supporting or contradicting the theoretical predictions.