2.3 Instantaneous speed and velocity

  Motion in One Dimension

The speed of the body at a particular instant of time is known as the instantaneous speed.

As we know, $$v = \frac{{\Delta s}}{{\Delta t}}$$
If the time interval $\Delta t$ is infinitesimally small such that it approaches zero $\left( {\Delta t \to 0} \right)$. Then the above ratio is called as instantaneous speed or referred as the speed of particle.
$${v_{inst}} = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta s}}{{\Delta t}}$$$${v_{inst}} = \frac{{ds}}{{dt}}$$

Instantaneous velocity is defined as the velocity of a particle at a particular instant of time.

As we know, $$\overrightarrow v = \frac{{\Delta \overrightarrow r }}{{\Delta t}}$$
Similarly, for instantaneous velocity $\left( {\Delta t \to 0} \right)$,
$$\overrightarrow v = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \overrightarrow r }}{{\Delta t}}$$$$\overrightarrow v = \frac{{d\overrightarrow r }}{{dt}}$$