Velocity Vector

In two or three dimensional situations the motion is not along a straight line. Consider an object which moves from a point ${{p}_{1}}$ to another point ${{p}_{2}}$ in two dimensional motion (xy-plane) as shown in Figure 4. In two or three dimensional motion the position is determined by the position vector.

Figure 4 The rate of change of displacement is velocity.

Let the position vector of point ${{p}_{1}}$ be ${\vec{r}_{1}}$ and that of ${{p}_{2}}$ be ${\vec{r}_{2}}$. Now the net displacement in the direction of motion is , \[\Delta \vec r = {{\vec{r}}_{2}}-{{\vec{r}}_{1}}\]

Suppose the object's initial time at point ${{p}_{1}}$ is ${{t}_{1}}$ and at ${{p}_{2}}$ is ${{t}_{2}}$. So, the total time during the motion between the points is $\Delta t={{t}_{2}}-{{t}_{1}}$. The velocity is the total displacement divided by the time taken during the displacement. Therefore the average velocity in vector form is,

\[{{\vec{v}}_{av}}=\frac{{{\vec{r}}_{2}}-{{\vec{r}}_{1}}}{{{t}_{2}}={{t}_{1}}}=\frac{{\Delta \vec r}}{\Delta t} \tag{3} \label{3}\]

The subscript indicates that the velocity is average velocity. When $\Delta t$ approaches zero the point ${{p}_{1}}$ moves closer and closer to the point ${{p}_{2}}$. Can you say the point ${{p}_{2}}$ moves closer and closer to the point ${{p}_{1}}$?

In this case $\Delta t$ approaches zero means that both ${{t}_{1}}$ and ${{t}_{2}}$ are almost equal, so we can also say that the point ${{p}_{2}}$ moves closer and closer to the point ${{p}_{1}}$ and vice versa. The instantaneous velocity in vector form in the limit that $\Delta t$ approaches zero is given as,

\[{{\vec{v}}_{\operatorname{ins}}}=\vec{v}=\frac{d\vec{r}}{dt}\]

The subscript "ins" is for the instantaneous velocity but we generally ignore this subscript and think the velocity as instantaneous velocity without that subscript. We also call the instantaneous velocity in vector form velocity vector.

Mechanics
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