Question

What is the importance of Instantaneous quantities?

Answer 1

The laws of physics are particularly simple when written using them. For example, the trajectory of a particle experiencing a force F can be predicted using Hamilton's Equations:
\[\frac{d}{dt} p(t) = F(x,t)\] and \[m\frac{d}{d t} x(t)= p(t)\] Those are pretty simple equations (although not so easy to solve, depending on F).