ï»¿ In Exercises 3.13 and 3.14, we have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why? from Physics Motion in Straight Line Class 11 Manipur Board

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In Exercises 3.13 and 3.14, we have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?

Instantaneous velocity is given by the first derivative of distance with respect to time.

i.e. ,Â
vInÂ = dxÂ /Â dtÂ

Here, the time intervalÂ dtÂ is so small that it is assumed that the particle does not change its direction of motion.

As a result, both the total path length and magnitude of displacement become equal is this interval of time.Â

Therefore, instantaneous speed is always equal to instantaneous velocity.
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Is absolute rest or absolute motion possible?

Absolute rest and absolute motion is not possible because there is no point in the universe which is at rest and can be taken as point of reference. The planet Earth is in continuous motion.
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What is motion?

When an object changes it's position with time, with respect to it's surroundings, the object is said to be in motion.
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Can a body be always at rest in some frame?

Yes, a body is always at rest in a frame which is fixed with the body itself.
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What is rest?

When an object does not change its position with time, with respect to it's surroundings. In such a case the object is said to be at rest.
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What is the most essential thing to study the motion of a body?

The most essential thing to study the motion is reference frame. Rest and motion are relative terms. It means that an object in one situation can be in rest but in another situation, the same object will be in motion.
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