Instantaneous Velocity Formula

In physics, you can take nothing for granted. Every concept needs to be built from ground up in the most fundamental of all sciences. Motion of objects is one such 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 basic concepts like instantaneous velocity, which are related to motion.

What is Instantaneous Velocity?

One of the attributes of velocity is that it is a vector. As you must already know, a vector is any physical quantity, whose complete 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.

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 velocity is 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

where ΔS is the total displacement and ΔT is the period for which displacement occurs.

The term instantaneous velocity identifies the speed attained by a particle or object, at a particular instant. Instead of talking about the average velocity attained over an extended period of time, with instantaneous velocity, I am only interested in the speed attained at any particular instant of motion.

Formula

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

Instantaneous Velocity = Lim_{ΔT → 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.

Calculation Technique

The calculation will be best illustrated by an example. Consider the following problem.

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

Solution: Instantaneous Velocity (at t = 5) = [dS/dt]_{t = 5} = [12t + 2]_{t = 5} = 62 m/s

With the knowledge, you can calculate the velocity of a 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.