In a motion of any particle usually, we have to calculate its velocity and speed in a given time period. But sometimes we are more interested in the value of the velocity at a specific point of time for a very small period of time. Instantaneous velocity is the type of velocity of an object in motion. This can be determined as the average velocity, but we may narrow the period of time so that it approaches zero. If a moving object has a standard velocity over a period of time, its average and instantaneous velocities may be the same. In this article, we will discuss the instantaneous velocity formula with examples. Let us begin learning!

## Instantaneous Velocity Formula

### Concept of Instantaneous Velocity

Instantaneous velocity is the rate of change of position for a time interval which is very small i.e. almost zero. It is measured using SI unit \( ms^{-1} \). Also, instantaneous speed is the magnitude of the Instantaneous velocity. It is having the same value as that of instantaneous velocity, with one difference that it does not have any direction.

In more simple words, the velocity of an object at that instant of time is known as instantaneous velocity. Thus, the definition is given as “The velocity of an object under motion at a specific point of time.”

If the object has uniform velocity then the instantaneous velocity may be the same as its standard velocity.

### The Formula for Instantaneous Velocity

It is computed as that of average velocity, but here time period is very much small. As we know that the average velocity for a given time interval is total displacement divided by the total time. As this time interval is tending towards zero, the displacement also approaches zero. But the limit of this ratio of displacement and time is non zero and we refer to it as instantaneous velocity.

Instantaneous Velocity Formula of the given body at any specific instant can be formulated as:

\( V_{int} = \lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t} = \frac {dx}{dt} \)

Wherewith respect to time t, x is the given function. We denote the Instantaneous Velocity in unit of \( ms^{-1}\) If any numerical contains the function of form f(x), we calculate the instantaneous velocity using the formula.

Where,

\( V_{int}\) | The instantaneous velocity of the body |

\( \Delta t \) | The small-time interval. |

x | The displacement variable |

t | The time |

It is a vector quantity. We can also determine it by taking the slope of the distance-time graph or x-t graph.

## Solved Examples on Instantaneous Velocity

Q.1: Find out the Instantaneous Velocity of a particle traveling along a straight line for time 3 seconds, with a position function x defined as 5² + 2t + 4?

Answer:

As given in the function,

x = 5t² + 2t + 4

Differentiating the given function with respect to t, we compute Instantaneous Velocity as follows:

\( V_{int} = \frac {dx}{dt} \)

Substituting function x,

\( V_{int} = \frac{d}{dt} 5t^2 + 2t + 4 \)

\( V_{int} = 10t+2 \)

Put value of t= 3, we get the instantaneous velocity as,

\(V_{int} = 10 \times 3 + 2 \)

\( V_{int} = 32 ms^{-1} \)

Thus the instantaneous velocity for the above function is \( 32 ms^{-1}\).

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