instantaneous velocity

Instantaneous velocity is your velocity right now, at this exact tick of the clock — the number a perfect speedometer with a direction arrow would read at a single instant. Average velocity blurs a whole trip into one figure; instantaneous velocity zooms all the way in to one moment and asks: how fast and in what direction am I moving at this precise instant?

Precisely, instantaneous velocity is the average velocity taken over a time interval that you shrink toward zero: v = limit of (delta x / delta t) as delta t approaches 0. In the language of calculus this limit is the derivative of position with respect to time, dx/dt. On a position-time graph, it is the slope of the tangent line — the line that just grazes the curve — at that instant. Its size is the instantaneous speed, and its SI unit is metres per second (m/s).

This is the velocity that shows up in the real laws of motion, because nature acts moment by moment, not trip by trip. The subtle point is that you cannot get it by dividing a finite distance by a finite time if the velocity is changing during that interval — that only gives an average. Instantaneous velocity is what that average approaches as the interval becomes vanishingly small, which is exactly why calculus was invented to handle motion.