Q.Define average velocity and instantaneous velocity. When are they the same?

[!TLDR] Average velocity is displacement over the whole interval divided by that interval; instantaneous velocity is its value at one instant, found by shrinking the interval to zero. [!ANSWER] $\vec{v}_{av}=\dfrac{\Delta \vec{r}}{\Delta t}$; $\vec{v}=\lim_{\Delta t\to 0}\dfrac{\Delta \vec{r}}{\Delta t}=\dfrac{d\vec{r}}{dt}$ — equal whenever the object moves with uniform (constant) velocity.

Average velocity over a time interval from $t_1$ to $t_2$ is the total displacement divided by that interval: $\vec{v}_{av} = \dfrac{\vec{r}_2-\vec{r}_1}{t_2-t_1} = \dfrac{\Delta\vec{r}}{\Delta t}$. Instantaneous velocity is the velocity at one specific instant of time, defined as the limiting value of average velocity as the averaging interval $\Delta t$ around that instant shrinks to zero: $\vec{v} = \lim_{\Delta t \to 0}\dfrac{\Delta \vec{r}}{\Delta t} = \dfrac{d\vec{r}}{dt}$. They are equal precisely when the velocity does not change over the interval being averaged — that is, whenever the object moves with uniform (constant) velocity, since then every sub-interval gives the same ratio $\Delta \vec{r}/\Delta t$, matching the instantaneous value at every point. [!ANSWER] $\vec{v}_{av}=\dfrac{\Delta \vec{r}}{\Delta t}$; $\vec{v}=\lim_{\Delta t\to 0}\dfrac{\Delta \vec{r}}{\Delta t}=\dfrac{d\vec{r}}{dt}$ — equal whenever the object moves with uniform (constant) velocity.

State the ratio-of-displacement definition for average velocity, then take its zero-interval limit for instantaneous velocity, and note when the two definitions coincide.

Assuming average and instantaneous velocity are always different — they coincide exactly whenever the velocity itself does not change during the interval considered.