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Linear Velocity

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The specification for the first Ferris wheel indicate that one trip around the wheel took 30 minute. How fast was the rider travelling around the wheel? The most spontaneous measure at which the rider is travelling around the wheel is what we call linear velocity. And in this section we will learn more about linear velocity.

Linear Velocity Definition

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The velocity or the linear velocity of an object is the time rate of change of linear displacement of the object. It is also defined as the speed of an object in a given direction.

The linear velocity can be of two types depending on the motion in which it is evolved:

Uniform Velocity: An object is said to be moving with a uniform velocity if it undergoes equal displacements in equal interval of time howsoever small these intervals may be.

Variable Velocity: An object is said to be moving with a variable velocity if it undergoes unequal displacements in equal intervals of time or equal displacements in unequal intervals of time or changes direction of motion while moving with a constant speed.

Linear Velocity Formula

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The linear velocity formula can be written as,

$Velocity$ = $\frac{Displacement}{Time\ interval}$

Velocity is a vector quantity, as it has both the magnitude (speed) and the direction. The velocity of an object can be positive, zero and negative according as its displacement is positive, zero or negative.

The unit of velocity is cms-1 in c.g.s. system and ms-1 in m.k.s. system or SI. The dimensional formula of velocity is [M0L1T-1].


The linear velocity can be calculated using the following formulae:

$V$ = $\frac{D}{T}$

Where,

$V$ = Linear velocity
$D$ = Displacement of the body
$T$ = the time taken to cover this displacement.
$v = u + at$

Where $v$ is the final linear velocity and $u$ is the initial linear velocity and ‘$a$’ is the acceleration of the body and $t$ is the time during which the motion is considered for calculations.

$v^{2} – u^{2} = 2aS$
Where $v$ and $u$ are the final and initial linear velocities respectively and ‘$a$’ is the acceleration of the body and $S$ is the displacement of the body.

Constant Linear Velocity

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If an object covers equal distances in equal intervals of time in a specified direction, then we say that the object is moving with constant linear velocity.

Average Linear Velocity

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Average linear velocity of a particle is refereed as the ratio of total linear displacement of the particle and the total time interval taken by the particle. It is denoted by $V^{av}$.

$V^{av}$ = $\frac{Total\ Displacement}{Total\ time\ taken}$

Angular Velocity to Linear Velocity

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There is a simple relation between the linear velocity and angular velocity for an object moving in an uniform circular motion. To arrive at the relationship, we first note that the displacement $\Delta s$ of an object moving through an angle $\Delta \theta$ around a circle of radius $r$ is the length of arc that subtends that angle at the center of the rotation.

Which is given as,

$\Delta s = r \Delta \theta$

Now divide both sides of the equation by the time interval $\Delta t$,

$\frac{\Delta s}{\Delta t}$ = $r$ $\frac{\Delta \theta}{\Delta t}$

But $\frac{\Delta s}{\Delta t}$ = $v$ and $\frac{\Delta \theta}{\Delta t}$ = $\omega$.

Hence,

$v = r \omega$

This formula gives us the linear velocity of an object moving in a circle of radius $r$ at angular velocity $\omega$.

Finding Linear Velocity

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The following are the linear velocity problems

Solved Examples

Question 1: A car starts travelling from rest and acquires a velocity of 2m/s2 in 20 seconds. Find the velocity after 20 sec.
Solution:
Given

$u=0$ 
$a=2$m/s2
$t= 20$ sec
$v$= ?

We have,

$v = u + at$,

$v = 0 + 2 \times 20$ = $40$ m/s.

The velocity of the car is $40$ m/s


Question 2: A car moves on a circular path with a linear velocity of 200 meters per second. If the particle makes 3 revolutions per second, find its angular velocity and also the radius of the circle?
Solution:
 
Angular velocity($\omega$) of the bus is = $3 \times 2\pi$ = $6\pi$

So the radius of the circle can be found from

$v = \omega \times r$

$200 = 6\pi \times r$

$r$ = $\frac{200}{6\pi}$

$r$ = $10.615$ meter 

So, the circular track’s radius is $10.615$ meters.