Instantaneous velocity is defined as the rate of change of position for a time interval which is very small (almost zero).

We know that the average velocity for a given time interval is total displacement divided by total time. As this time interval approaches zero, the displacement also approaches zero. But the limit of the ratio of displacement and time is non zero and is called the instantaneous velocity.

If the displacement of the particle varies with respect to time and is given as (6t^{2} + 2t + 4) m, the instantaneous velocity can be found out at any given time by:

s = (6t^{2} + 2t + 4)

Velocity (v) = \( \frac {ds}{dt} \)

= \( \frac{d(6t^2 + 2t + 4)}{dt}\)

= 12t + 2

So if we have to find out the instantaneous velocity at t = 5sec, then we will put the value of t in the obtained expression of velocity.

Instantaneous velocity at t = 5 sec = (12×5 + 2) = 62 m/s

Let us calculate the average velocity now for 5 seconds now.

Displacement = (6×5^{2} + 2×5 + 4) = 164 m

Average velocity = \( \frac{164}{5}\)

### Instantaneous Speed:

We know that the average speed is for a given time interval is total distance travelled divided by the total time taken. As this time interval approaches zero, the distance travelled also approaches zero. But the limit of the ratio of distance and time is non zero and is called the instantaneous speed.

To understand it in simple words we can also say that instantaneous speed at any given time is the magnitude of instantaneous velocity at that time.

If distance as a function of time is known to us, we can find out the instantaneous speed at any time. Let’s understand this by the means of an example.

Distance (s) = 5t^{3} m

Speed (v) = \( \frac {ds}{dt} \)

= \( \frac{d(5^{t^3})}{dt}\)

= 15t^{2}

We can now easily find the instantaneous speed at any given time by putting the value of t in this obtained expression.

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