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# Instantaneous Speed and Velocity

Let’s carry out a small activity. You are walking towards theatre from home. Suddenly you realize that you are running late and would probably miss the beginning. You start running pretty fast. Maybe it’s been a while since the last time you ran and you slow down a bit. This is ‘instantaneous speed’.  And if you include the direction with that speed then you get the instantaneous velocity.

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## Instantaneous velocity

The rate of change of displacement of an object in a particular direction is its velocity. Its S.I unit is meter per second.
$$\vec{v} = \frac{d \vec{s}}{dt}$$
The direction of instantaneous velocity at any time gives the direction of motion of a particle at that point in time. The magnitude of instantaneous velocity equals the instantaneous speed. This happens because, for an infinitesimally small time interval, the motion of a particle can be approximated to be uniform.

## Instantaneous Speed

The average velocity tells us how fast an object has been moving over a given time interval but does not tell us how fast it moves at different instants of time during that interval. For this, we define instantaneous speed. It is the rate of change of distance with respect to time.

$$v = \frac{ds}{dt}$$

Instantaneous speed is always greater than or equal to zero and is a scalar quantity. For uniform motion, instantaneous speed is constant. To understand it in simple words we can say that instantaneous speed at any given time is the magnitude of instantaneous velocity at that time. It is a limit of the average speed as the time interval become very small.

A moving object does not have the same speed during its travel. Sometimes it speeds up and sometimes slows down. At a given instant time what we read from the speedometer is instantaneous speed. When a cop pulls you over for speeding, he clocked your car’s instantaneous speed or speed at a specific point in time as your car sped down the road.

## Solved Examples For You

Q.Which of the following sentences is an example of instantaneous velocity?

1. The car covered 500 kilometers first 10 hours of its northward journey.
2. Five seconds into launch, the rocket was shooting upward at 5000 meters per second.
3. The Cheetah can run at 70 miles per hour.
4. Moving at 5 kilometres per hour, it will take us eight hours to get to the base.
5. Roger Bannister was the first person to run one mile in less than four minutes.

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