Velocity

Velocity of an object is defined as a time rate of change of displacement of the object. It is also define as the speed of an object in a given direction. Quantitatively,

$ \displaystyle \text{velocity = }\dfrac{{\text{displacement}}}{{\text{time interval}}}$

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

The unit of velocity is cm/s in CGS system and m/s in MKS system or S.I. The dimensional formula of velocity is $ \displaystyle \left[ {{{M}^{0}}{{L}^{1}}{{T}^{{-1}}}} \right]$.

1. Uniform velocity

An object is said to be moving with a uniform velocity if it undergoes equal displacement in equal intervals of time howsoever small this intervals maybe.

2. Variable velocity

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

3. Average velocity

It is that uniform velocity with which the object will cover the same displacement in same interval of time as it does with its actual variable velocity during that time interval.

4. Instantaneous velocity

The average velocity of a particle during a time interval cannot tell us how fast, or in what direction, the particle was moving at any given time during the interval. To describe the motion in greater detail we need to define the velocity at any specific instant of time or specific point along the path. Such a velocity is called instantaneous velocity it needs to be defined carefully.

We know that average velocity is the ratio of total displacement upon total time taken i.e.

$ \displaystyle {{\overrightarrow{v}}_{{av}}}=\dfrac{{\text{total displacement}}}{{\text{total time taken}}}$

Instantaneous velocity is the limit of the average velocity as the time interval approaches zero; it equals the instantaneous rate of change of position with time. We use the symbol $ \displaystyle v$ with no subscript for instantaneous velocity:

$ \displaystyle v=\underset{{\Delta t\to 0}}{\mathop{{\lim }}}\,\dfrac{{\Delta x}}{{\Delta t}}=\dfrac{{dx}}{{dt}}$

We always assume that the time interval $ \displaystyle {\Delta t}$ is positive so that $ \displaystyle v$ has the same algebraic sign as $ \displaystyle {\Delta x}$

If the positive x-axis points to the right, a positive value of $ \displaystyle v$ means that $ \displaystyle x$ is increasing and the motion is towards the right; negative value of $ \displaystyle v$ means that $ \displaystyle x$ is decreasing and the motion is towards the left. A body can have positive $ \displaystyle x$ and negative $ \displaystyle v$ are the reverse; $ \displaystyle x$ tells us where the body is, while $ \displaystyle v$ tell us how its moving.

Instantaneous velocity, like average velocity is a vector quantity.