Assertion : The instantaneous velocity is given by the limiting value of the average velocity as the

Question

Assertion : The instantaneous velocity is given by the limiting value of the average velocity as the time interval approaches zero.\ Reason : The direction of the average velocity is same as that of displacement.

Professor PC Thomas Classes · Answer

(b) : The average velocity $|\vec{v}|$ of an object is the ratio of the net displacement and the corresponding time interval. \ $ \vec{v}=\frac{\Delta \vec{r}}{\Delta t}=\frac{\Delta x \hat{i}+\Delta y \hat{j}}{\Delta t}=\hat{i} \frac{\Delta x}{\Delta t}+\hat{j} \frac{\Delta y}{\Delta t} $ \  or $\vec{v}=v_x \hat{i}+v_y \hat{j}$ Since $\vec{v}=\frac{\Delta \vec{r}}{\Delta t}$, the direction of the average velocity is the same as that of $\Delta \vec{r}$. The instantaneous velocity is the limiting value of the average velocity as the time interval approaches zero. \ $ 	herefore \quad \vec{v}=\lim _{\Delta t ightarrow 0} \frac{\Delta \vec{r}}{\Delta t}=\frac{d \vec{r}}{d t} . $