Distance, Displacement, Speed And Velocity

VELOCITY

The displacement of a body in unit is called it's Velocity.
It is a vector quantity.
CGS unit is \( cms^{ - 1} \)
SI unit is \( ms^{ - 1} \).
Dimensional formula is  \( [L^{ - 1} T^{ - 1} ] \)

Distance, Displacement, Speed And Velocity

UNIFORM VELOCITY
If a body has equal displacements in equal intervals of time however small the intervals may the then it is said to be moving with uniform velocity. The motion is expressed by an equation of the form 
                x=vt+b

Where x is the position coordinate of the particle, t is the time, v and b are certain constants. In this equation position is a linear function of time. Hence the position - time graph is a straight line, the slope  of which is a constant(v) and is equal to uniform velocity of particle.
                 
   
 Position - time graph of an object in uniform motion 
Where t=0,the above equation can be written \( x_{_0 } = b \)
Where \( x_{_0 } \) indicates the initial position of the particle from the origin.

NON-UNIFORM VELOCITY
If a body has equal displacements in unequal intervals of time or unequal displacements in equal intervals of time then it is said to be moving with Non-Uniform Velocity.

Note : The displacement variation may be due to change in magnitude or change in direction of motion or both.

Distance, Displacement, Speed And Velocity

AVERAGE VELOCITY
For a particle in motion (uniform or non- uniform), the ratio of total displacement to the total time interval is called Average velocity.
                         \( \text{Average velocity } = \frac{{\text{Total displacement}}} {{\text{Total time}}} \)
Suppose a particle displaces from P₁ to P₂, in a time interval \(\Delta t\). If  x₁ is initial position and x₂ is final position then
Average velocity = (v) \( = \frac{{x_2 - x_1 }} {{\Delta t}} \)

INSTANTANEOUS VELOCITY
The velocity of a particle at a particular instant of time is called it's instantaneous velocity. (or) It is also defined as the limit of average velocity as the time interval (\(\Delta t\)) becomes infinitesimally small.
If  \(\Delta s\) is the displacement by a particle in a time interval \(\Delta t\) then
            Velocity=V=\( \frac{{\Delta S}} {{\Delta t}} \)
If the time interval \(\Delta t\) is chosen to be very small, i.e., as \(\Delta t \to 0\), the corresponding velocity is called instantaneous velocity.
        \( \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta S}} {{\Delta t}} = \frac{{dS}} {{dt}} \)=instantaneous velocity
The instantaneous velocity is rate of change of position with time.
The velocity at a particular instant is equal to the slope of the tangent drawn on position time graph at that instant.