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# Average Velocity

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 Sub Topics Before going to the concept of average velocity, one should know about the concept of velocity. Velocity - the rate of change of position of an object. It is a vector quantity because it requires both magnitude and direction to define. Consider a body moving with different velocities throughout its journey. The magnitude of its velocity changes with time. To find the velocity of the body on the whole of the journey of the body, we use the average velocity concept. In this section we will learn more about average velocity.

## Average Velocity Definition

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

If '$s$' is total the displacement of the body and '$t$' is the total time taken by the body to complete the displacement, we have $V_{ave}$ = $\frac{s}{t}$.

## Average Velocity Formula

Average velocity of a particle is refereed as the total distance displacement of the particle and the total time interval taken by the particle. It is denoted by $V_{ave}$.

$V_{ave}$ = $\frac{Total\ Displacement}{Time\ Interval}$

If the particle is travelled a distance of $\Delta s$ in the time interval $\Delta t$, then the average velocity of a particle is

$V_{ave}$ = $\frac{\Delta s}{\Delta t}$ = $\frac{r_{2}-r_{1}}{t_{2}-t_{1}}$

1. If the body reaches the initial position at the end of its journey, the displacement of the body is zero and hence its average velocity is zero. For example, if a stone is projected vertically upwards from ground, it reaches the same position at the end of its time of flight. In the process, the total displacement of the body is zero. Hence, in this case, the average velocity of the body is zero.
2. When a body moves in a circular path, at the end of a revolution in the circular path, the displacement of the body is zero, Hence the average velocity of the body is zero.

## Average Speed vs Average Velocity

Average of Speeds are called the average speed. It is not a Vector Quantity. If there are many distances travelled proportionate to the different time taken, then we need to determine the Average Speed.

$Average\ Speed$ = $\frac{Total\ Distance\ travelled}{Total\ time\ taken}$

Average velocity is the average of velocities. Average velocity is a Vector Quantity. It got both magnitude and direction. When there are displacements corresponding to the different time taken, then we need to determine the Average velocity.

$Average\ Speed$ = $\frac{Total\ Displacement\ travelled}{Total\ time\ taken}$

## Magnitude of Average Velocity

Magnitude of the average velocity is defined as the ratio of the distance travelled by the body to the time taken. In general, if the body covers distance $d_{1}, d_{2}, d_{3},....d_{n}$ in time intervals $t_{1}, t_{1}, t_{1}....t_{1}$, then the average velocity is given by

$v_{ave}$ = $\frac{\sum_{i = 1}^{n}d_{i}}{\sum_{i=1}^{n}t_{i}}$

The direction of average velocity vector is the same as that of the displacement vector.

## Finding Average Velocity

The following are the problems of average velocity.

### Solved Examples

Question 1: A particle moving along the x axis is located at 17.2 m at 1.79 s and at 4.48 m at 4.28 s. what is the average velocity in the particular time interval?
Solution:
Distance traveled by particle beside x-axis = Final location  - Initial location

= 17.20 - 4.48

= 12.72 meters

Time taken to cover up this distance = Final time - Initial time

= 4.28 - 1.79

= 2.49 seconds

Average velocity    = (Distance traveled)/(Time taken)

= 12.72 / 2.48

= 5.1084 (approximately)

Average velocity of the particle is    5.1084 m/s

Question 2: A boy runs 100 metres in 50 seconds in going from his home to the shop in the East direction, and then runs a distance of 100 metres again in 50 seconds in the reverse direction from shop to reach home. Calculate average velocity.
Solution:

We have $V_{ave}$ = $\frac{Total\ Displacement}{Time\ Interval}$

The boys runs East 100 m and then 100m in opposite direction( towards West ) .

Total displacement = $100$ m - $100$ m = $0$m

Total time taken = $50$ sec + $50$ sec = $100$ sec

$V_{ave}$ = $\frac{Total\ Displacement}{Time\ Interval}$

$V_{ave}$ = $\frac{0}{100}$ = $0$ m/s