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Instantaneous Velocity Formula

Omkar Phatak Apr 20, 2019
Instantaneous velocity, as the name suggests, refers to the velocity at a particular instance. This write-up highlights the definition of instantaneous velocity, equation, and a problem along with its solution.
In physics, motion of objects is a very important concept. The various notions related to the concept of motion, including frame of reference, displacement, velocity, and momentum need to be defined.
Before you can attack the complex problems of physics, it is necessary that one gets a good grounding in the basic concepts like instantaneous or average velocity, which are related to motion.
Instantaneous and average velocities can be different if the velocity is not maintained constant. These terms can be best explained with the example of sprinters running a race.
Their velocities are varying as they start and end the race. The value of velocity of at each instance of time on the running track is instantaneous, while the average of all values indicates the average velocity.

What is Instantaneous Velocity?

One of the attributes of velocity is that it is a vector quantity. As we all know, a vector is any physical quantity whose description is incomplete without a specification of direction associated with it.
Velocity can be defined as the distance traveled or displacement made by an object per unit time in a specific direction. It could also be defined in other words as rate of change in position of an object in unit time. Speed is purely the magnitude of velocity and doesn't take direction into account.
Whenever an object moves, it does so at a certain velocity. Motion of an object, and hence velocity, is relative to a frame of reference. Normally, what we call as velocity is the average velocity. It is the average speed maintained by an object over a specific period of time in a particular direction.
Average velocity (V) can be defined by the following formula:

Average Velocity (V) = Displacement ÷ Time = ΔS ÷ ΔT

ΔS is the total displacement
ΔT is the period for which displacement occurs
The term instantaneous velocity represents the speed attained by a particle or an object, at a particular instant of time. Instead of considering the average velocity attained over an extended period of time, it is beneficial to consider the speed attained at any particular instant of motion.

Formula for Instantaneous Velocity

The concept of an instance is best understood mathematically in terms of a limit. Here is the formula in terms of a limit:

Instantaneous Velocity = LimΔT → 0 ΔS / ΔT = dS / dT

The value at any particular moment of motion can be calculated by taking the derivative of equation of motion for displacement, and substituting the value of time in the result.

Calculating Instantaneous Velocity

In order to calculate the instantaneous velocity of an object, we need to know its displacement and the time required.


The equation of motion for a particle (in motion) is 6t2 + 2t + 1. Find the instantaneous velocity of the particle at t = 5s.


Instantaneous Velocity (at t = 5) = [dS ÷ dt]t = 5 = [12t + 2]t = 5 = 62 m/s
With this knowledge, you can calculate the velocity of any particle or object in classical physics, if you know the equation of motion. In fact, Newtonian mechanics―the knowledge of the equation of motion―can provide you with all the answers.