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

The average velocity $(\vec{v})$ of an object is the ratio of the displacement and the corresponding time interval.

Suppose, the object covers $\Delta \vec{r}$ displacement in time interval $\Delta t$.

Average velocity,

$\langle\vec{v}\rangle=\frac{\overrightarrow{\Delta r}}{\Delta t}=\frac{\Delta x \hat{i}+\Delta y \hat{j}}{\Delta t}=\hat{i}\left(\frac{\Delta x}{\Delta t}\right)+\hat{j}\left(\frac{\Delta y}{\Delta t}\right)$

or $\langle\vec{v}\rangle=\left\langle v_{x}\right\rangle \hat{i}+\left\langle v_{y}\right\rangle \hat{j}$

Since $\langle\vec{v}\rangle=\frac{\Delta \vec{r}}{\Delta t}$, the direction of the average velocity is the same as that of $\Delta \vec{r}$

Suppose, the object covers $\Delta \vec{r}$ displacement in time interval $\Delta t$

Average velocity, $\langle\vec{v}\rangle=\frac{\Delta \vec{r}}{\Delta t} \quad \ldots$ $(1)$

After times $\Delta t_{1}, \Delta t_{2}$ and $\Delta t_{3} \cdot \overrightarrow{\Delta r_{1}}, \overrightarrow{\Delta r}_{2}$ and $\overrightarrow{\Delta r}_{3}$ are the displacements of the object in time $\Delta t_{1}$, $\Delta t_{2}$ and $\Delta t_{3}$ respectively.

Instantaneous velocity is given by the limiting value of the average velocity as the time interval approaches zero.

$\left(\vec{v}=\lim _{\Delta t \rightarrow 0} \frac{\overrightarrow{\Delta r}}{\Delta t}=\frac{\overrightarrow{d r}}{d t}\right)$

The direction of velocity at any point on the path of an object is tangential to the path at that point and is in the direction of motion.

The units of average velocity and instantaneous velocity in MKS system is $\frac{\mathrm{m}}{\mathrm{s}}$ and in CGS system is $\frac{\mathrm{cm}}{\mathrm{s}}$.

Average velocity,

$\langle\vec{v}\rangle=\frac{\Delta \vec{r}}{\Delta t}$

$\therefore\langle\vec{v}\rangle=\frac{\Delta x}{\Delta t} \hat{i}+\frac{\Delta y}{\Delta t} \hat{j}$

$\langle\Delta \vec{v}\rangle=\frac{\Delta x}{\Delta t} \hat{i}+\frac{\Delta y}{\Delta t} \hat{j}$

By taking $\lim _{\Delta t \rightarrow 0}$ instantaneous velocity is obtained.

$\vec{v}=\hat{i}\left(\frac{d x}{d t}\right)+\hat{j}\left(\frac{d y}{d t}\right)-v_{x} \hat{i}+v_{y} \hat{j}$

where $v_{x}=\frac{d x}{d t}, v_{y}=\frac{d y}{d t}$

So, if the expression for the coordinates $x$ and $y$ are known as functions of time, we can use these equations to find $v_{x}$ and $v_{y}$

The magnitude of $\vec{v}$ is then,

$v=\sqrt{v_{x}^{2}+v_{y}^{2}}$