Physics - Kinematics Question with Solution | TestHub

Let the particle 'P' move along a curve between two positions, 1 and 2, at times $t=t_1$ and $t=t_2$, respectively. During the time interval $\Delta t=(t_2-t_1)$, the position vector of the particle changes from ${\vec{r}}_1$ to ${\vec{r}}_2$ by $\Delta \vec{r}$. Hence, on average, the position vector changes with time at a rate:

$${\vec{v}}_{av}=\frac{\Delta \vec{r}}{\Delta t}=\frac{{\vec{r}}_2-{\vec{r}}_1}{t_2-t_1}$$

For the instantaneous velocity, as $\Delta t\to 0$:

$$\vec{v}=\underset{\Delta t\to 0}{\lim }{\vec{v}}_{av}=\underset{\Delta t\to 0}{\lim }\frac{\Delta \vec{r}}{\Delta t}=\frac{d\vec{r}}{dt}$$