# 1.14 Velocity

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Velocity determines motion of an object at a given instant in both magnitude and direction. An object can have only one velocity at a given time. It is not possible for an object to have two velocities at the same time.

Velocity is the measure of rapidity with which a particle covers shortest distance between initial and final positions, irrespective of the actual path. It also indicates the direction of motion as against speed, which is devoid of this information.

Velocity
Velocity is the rate of change of displacement with respect to time and is expressed as the ratio of displacement and time.

$\begin{array}{l}\mathbf{v}=\frac{\mathrm{Displacement}}{\mathrm{\Delta t}}\end{array}$

$\begin{array}{l}⇒ \mathrm{Displacement}=\mathbf{\Delta v}t\end{array}$

If the ratio of displacement and time is evaluated for finite time interval, we call the ratio “average” velocity, whereas if the ratio is evaluated for infinitesimally small time interval(Δt→0) , then we call the ratio “instantaneous” velocity. Conventionally, we denote average and instantaneous velocities as ${\mathbf{v}}_{\mathbf{a}}$ and $\mathbf{v}$ respectively to differentiate between the two concepts of velocity.

As against speed, which is defined in terms of distance, velocity is defined in terms of displacement. Velocity amounts to be equal to the multiplication of a scalar (1/Δt) with a vector (displacement). As scalar multiplication of a vector is another vector, velocity is a vector quantity, having both magnitude and direction. The direction of velocity is same as that of displacement and the magnitude of velocity is numerically equal to the absolute value of the velocity vector, denoted by the corresponding non bold face counterpart of the symbol.

Dimension of velocity is $L{T}^{-1}$    and its SI unit is meter/second (m/s).

## Position vector and velocity

The displacement is equal to the difference of position vectors between initial and final positions. As such, velocity can be conveniently expressed in terms of position vectors.

Let us consider that ${\mathbf{r}}_{\mathbf{1}}$ and ${\mathbf{r}}_{\mathbf{2}}$ be the position vectors corresponding to the object positions at time instants ${t}_{1}$ and ${t}_{2}$ . Then, displacement is given by :

$\begin{array}{lll}\mathbf{v}=& \frac{{\mathbf{r}}_{\mathbf{2}}-{\mathbf{r}}_{\mathbf{1}}}{{t}_{2}-{t}_{1}}& =\frac{\Delta \mathbf{r}}{\Delta t}\end{array}$

Velocity
Velocity is the rate of change of position vector with respect to time and is expressed as the ratio of change in position vector and time.

The expression of velocity in terms of position vectors is generally considered more intuitive and basic to the one expressed in terms of displacement. This follows from the fact that displacement vector itself is equal to the difference in position vectors between final and initial positions.

## Average velocity

Average velocity is defined as the ratio of total displacement and time interval.

$\begin{array}{ll}{\mathbf{v}}_{a}=\frac{\Delta \mathbf{r}}{\Delta t}& =\frac{{\mathbf{r}}_{\mathbf{2}}-{\mathbf{r}}_{\mathbf{1}}}{{t}_{2}-{t}_{1}}\end{array}$

Average velocity gives the overall picture about the motion. The magnitude of the average velocity tells us the rapidity with which the object approaches final point along the straight line – not the rapidity along the actual path of motion. It is important to notice here that the magnitude of average velocity does not depend on the actual path as in the case of speed, but depends on the shortest path between two points represented by the straight line joining the two ends. Further, the direction of average velocity is from the initial to final position along the straight line (See Figure).

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