# 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).

#### Questions & Answers

List the application of projectile
Luther Reply
How can we take advantage of our knowledge about motion?
Kenneth Reply
pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
Mercy Reply
what are scalars
Abdhool Reply
show that 1w= 10^7ergs^-1
Lawrence Reply
what's lamin's theorems and it's mathematics representative
Yusuf Reply
if the wavelength is double,what is the frequency of the wave
Ekanem Reply
What are the system of units
Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
58asagravitasnal firce
Amar
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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