Question

We have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?

Answer 1

Instantaneous velocity is given by the first derivative of distance to time i.e.
$V_{in}=\frac{dv}{dt}$
Here, the time interval dt is so small that it is assumed that the particle does not change its direction of motion. As a result, both the total path length and magnitude of displacement become equal over time. Therefore, instantaneous speed is always equal to instantaneous velocity.