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Velocity

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When one know both the speed and the direction of an object, we know its velocity.

For example, if a bus travels at 55 km/h, we know its speed. But if we say it moves at 55 km/h to the north, we specify its velocity. Speed is a description of how fast; velocity is how fast and in what direction.  In this section we will learn more about velocity.
The velocity of an object is the change in its position, divided by the time it took for this change to occur. Velocity is a vector and has both a magnitude and a direction. 

Formula for Velocity

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Mathematically, the velocity of an object is

$\vec{v}$ = $\frac{\Delta x}{\Delta t}$

Where $\vec{v}$ is the velocity vector, $\delta x$ is the displacement vector and $\Delta t$ is the time interval over which the displacement occurs. The change in the position of an object is its final position minus its initial position.

i.e., $\Delta x$ = $x_{f} – x_{i}$

The equation for velocity can be written as

$v$ = $\frac{Final\ position – Initial\ position}{time}$ =$\frac{x_{f} – x_{i}}{t}$
Velocity units: Velocity and speed are expressed in metre per sec, ie., $ms^{-1}$. The dimensional formula is [$LT^{-1}$].
The initial velocity of a substance is the velocity it will be moving at time t = 0. The final velocity of the substance is the velocity of the substance that will be moving at time t. Both final and initial velocity can be zero at certain situation.

If a car starts from rest, its initial velocity is zero. If a projectile is tossed into the space, its initial velocity will be more than zero. If a car stops after applying the brake, the initial velocity will be more than zero, but the final velocity will be zero. Generally the initial velocity is denoted by ‘u’ and the final velocity is denoted by ‘v.’

In one dimensional motion, it has more importance to solve the displacement(s), velocity (u or v) and acceleration (a).
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Uniform or constant velocity is nothing but when a body travels in a straight line and covers equal distance in equal interval of time; it is said to have constant velocity.

For example, a bike traveling with a constant speed in a straight line has uniform velocity.
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If the velocity of a body in a particular direction changes continuously at a uniform rate, then the mean of the initial and final velocity over a given period of time is called the average velocity in that direction.

$Average\ velocity$ = $\frac{Initial\ velocity\ + Final\ velocity}{2}$ = $\frac{u+v}{2}$

Where $u$ is the initial velocity and $v$ is the final velocity in a particular direction.
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Instantaneous Velocity

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This is the velocity of a body at any particular instant during its motion. This can be obtained by dividing the distance travelled in a very short interval of time.

$Instantaneous\ velocity$ = $\frac{Distance\ travelled\ in\ a\ very\ short\ interval}{Time interval}$
Linear velocity $v$ is defined as the rate of change of linear displacement $s$ with respect to time $t$. The unit of linear velocity is m/s.

For motion in a straight line,

$Linear\ velocity$ = $\frac{Change\ of\ displacement}{Change\ of\ time}$
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Angular Velocity 

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Angular velocity is defined as the rate of change of angular displacement (\theta) with respect to time (t).

$Angular\ velocity $ = $\frac{Angle\ turn\ through}{Time\ taken}$

$\omega$ = $\frac{\theta}{t}$

The speed of revolution of a wheel or a shaft is usually measured in revolution per minute or revolution per second. The unit of angular velocity is radian per second.

Velocity and Speed

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Speed Velocity
  1. The speed of a body is defined as the distance traveled by it per unit time.
  2. It is a scalar quantity.
  3. The value of speed can be positive or zero
  1. The velocity of a body is defined as the distance travelled by it per unit time in a given direction.
  2. It is a vector quantity
  3. The value of speed can be positive, negative or zero

Examples of Velocity

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The following are the examples of velocity.

Solved Examples

Question 1: Electron moving along the x direction is situated at 28.4 m at 2.89 s and at 6.23 m at 6.28 s. Find the velocity over the time interval?
Solution:
 
We have, 

$Velocity (v$) = $\frac{Change\ of\ displacement}{Change\ of\ time}$ 

Change of Displacement = Final position  - Initial position 

= 28.4 m – 6.23 m

= 22.17 meters

Change of time = Final time - Initial time

= 6.28 – 2.89

= 3.39 seconds

$Velocity (v$) = $\frac{Change\ of\ displacement}{Change\ of\ time}$ 

= $\frac{22.17}{3.39}$ 

= 6.54 (approximately)

Velocity of the electron is 6.54 m/s
 

Question 2: In a 1000 meter race, the winner takes 20 seconds to reach the finishing point. Calculate the velocity of the winner.
Solution:
 
Given,

Displacement = 1000 m

Time= 20 s

We know that,

$Linear\ velocity$ = $\frac{Change\ of\ displacement}{Change\ of\ time}$
 
Then the velocity of the winner = $\frac{Change\ of\ displacement}{Change\ of\ time}$

                           = $\frac{1000}{20}$

                           = $50$ m/s   

The velocity of the winner is $50$ m/s