The everyday concept of velocity arises when we consider how fast or slow a body moves. Somehow, we relate the displacement of the body with the time spent in such displacement. This type of relationship is expressed in the form of instantaneous speed. In this, we will learn about instantaneous speed and the instantaneous speed formula.

## Â **What Is Speed?**

The rate of change of position of any object with time is known as speed. The speed of an object can change as it moves. The speed of an object at a certain instant of time is known as the instantaneous speed. If the position is a function of time, then the speed depends on the change in the position as time changes.

The instantaneous speed can be found as this change in time becomes small. Calculating the instantaneous speed requires finding the limit of the position function as the change in time approaches zero.

## Instantaneous Speed

It is the rate of change of distance of an object with respect to time. The unit for speed is meters per second (m/s).

**Instantaneous speed (v) = distance/ time**

v = limit as change in time approaches zero \(\frac{change in position}{change in time}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0} \frac{[x(t+\Delta t)]-x(t)}{\Delta t}\)

Where,

v | instantaneous speed (m/s) |

\(\Delta\) | change in values |

x | displacement (m) |

t | time (s) |

Instantaneous speed is always greater than or equal to zero. Instantaneous speed is a scalarÂ quantity. For uniform motion, instantaneous speed is constant. In other words, we can say that instantaneous speed at any given time is the magnitude of instantaneous velocity at that time. Instantaneous speed is a limit of theÂ average speedÂ as the time interval become very small.

**Solved Example forÂ Instantaneous Speed Formula**

Q. 1: When an object is dropped and acted on by gravity, its position changes according to the function x(t) = 4.9t^{2}, and x(t) is in units of meters. What is the instantaneous speed at t = 10.0 sec usingÂ Instantaneous Speed Formula?

Answer:Â The instantaneous speed formula:

\(v = \lim_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0} \frac{[x(t+\Delta t)]-x(t)}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{[4.9(t + \Delta t)^2Â ]â€“ 4.9t^2}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{[4.9(t^2Â + 2t\Delta t + \Delta t^2)]â€“ 4.9t^2}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{4.9t^2Â + 9.8t\Delta t + 4.9\Delta t^2Â â€“ 4.9t^2}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{9.8t\Delta t + 4.9\Delta t^2}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0} (9.8t + 4.9\Delta t)\)

= 9.8t

= 9.8(10.0) m/s

= 98.0 m/s

The instantaneous speed at t = 10.0 sec is 98.0 m/s.

Q. 2: A car stops at a traffic light, and then begins moving along a straight road. The car’s distance from the light is given by the function x(t) = 6t2. What will be the instantaneous speed at t = 5.00 s?

Answer:Â The instantaneous speed formula:

\(v = \lim_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0} \frac{[x(t+\Delta t)]-x(t)}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{[6(t + \Delta t^2Â ]â€“ 6t^2}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{[6(t^2Â + 2t\Delta t + \Delta t^2)]â€“ 6t^2}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{6t^2Â + 12t\Delta t + 6\Delta t^2Â â€“ 6t^2}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0}\frac{12t\Delta t + 6\Delta t^2}{\Delta t}\)

\(= \lim_{\Delta t\rightarrow 0} (12t + 6\Delta t)\)

= 12t

= 12(5.0) m/s

= 60.0 m/s

The instantaneous speed of the car at t = 5.00 s is 60 m/s.

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