Explain average velocity,instantaneous velocity,and components of velocity for motion in a plane.

(N/A) The average velocity $(\vec{v})$ of an object is the ratio of the displacement and the corresponding time interval.
Suppose an object covers a displacement $\Delta \vec{r}$ in a time interval $\Delta t$.
Average velocity is given by:
$\langle\vec{v}\rangle = \frac{\Delta \vec{r}}{\Delta t} = \frac{\Delta x \hat{i} + \Delta y \hat{j}}{\Delta t} = \hat{i} \left( \frac{\Delta x}{\Delta t} \right) + \hat{j} \left( \frac{\Delta y}{\Delta t} \right)$
Or,$\langle\vec{v}\rangle = \langle v_{x} \rangle \hat{i} + \langle v_{y} \rangle \hat{j}$
The direction of the average velocity is the same as that of the displacement vector $\Delta \vec{r}$.
Instantaneous velocity is defined as the limiting value of the average velocity as the time interval approaches zero:
$\vec{v} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$
The direction of velocity at any point on the path is tangential to the path at that point and is in the direction of motion.
In terms of components,the instantaneous velocity is:
$\vec{v} = \hat{i} \left( \frac{dx}{dt} \right) + \hat{j} \left( \frac{dy}{dt} \right) = v_{x} \hat{i} + v_{y} \hat{j}$
where $v_{x} = \frac{dx}{dt}$ and $v_{y} = \frac{dy}{dt}$ are the components of velocity along the $x$ and $y$ axes respectively.
The magnitude of the velocity vector is given by:
$v = \sqrt{v_{x}^{2} + v_{y}^{2}}$
The units of velocity in the $MKS$ system are $m/s$ and in the $CGS$ system are $cm/s$.