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What is the definition of speed and velocity?

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The speed (or the average speed) is defined as the total distance traveled divided by the total time interval during which the motion has taken place.What is the definition of speed and velocity?

The velocity (or the average velocity) is defined as the change in position (or displacement) divided by the time intervals in which the displacement occurs.

Thread 'Average speed and average velocity' https://www.physicsforums.com/threads/average-speed-and-average-velocity.986630/

Now consider the motion of an object along a straight line, the magnitude of the displacement is equal to the total distance. In this case, the magnitude of average velocity is equal to the average speed. But if the motion involves change in direction then the distance is greater than the magnitude of displacement. Thus, in this case the average speed is not equal to the magnitude of the average velocity.

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Any good maths student could construct an example where the limits do not agree. Generally, this will involve so-calledNow consider the motion of an object along a straight line, the magnitude of the displacement is equal to the total distance. In this case, the magnitude of average velocity is equal to the average speed. But if the motion involves change in direction then the distance is greater than the magnitude of displacement. Thus, in this case the average speed is not equal to the magnitude of the average velocity.

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Right. And which is larger, the displacement or the distance?Thus, in this case the average speed is not equal to the magnitude of the average velocity.

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Distance is greater than or equal to to the magnitude of the displacement... I have understood the distinction between the magnitude of average velocity over an interval of time and the average speed over the same interval. The average speed it is defined as the total distance divided by time and not as magnitude of average velocity.Right. And which is larger, the displacement or the distance?

But then why is the instantaneous speed always equal to the magnitude of instantaneous velocity?

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Can you please explain a little bit more about the pathological functions that you mentioned? seems interesting, and I would like to learn more about it.Any good maths student could construct an example where the limits do not agree. Generally, this will involve so-calledpathologicalfunctions, which are not possible as a physical trajectory. E.g. some sort of infinite oscillation.

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You can approximate a curve as a series of straight lines. More straight lines is a better approximation. Infinitely many infinitely short straight lines is a perfect approximation - so at an instant there is no difference between following a curve and following an infinitely short straight line (the tangent to the line, in fact).But then why is the instantaneous speed always equal to the magnitude of instantaneous velocity?

If I teleport from one point to another then I didn't follow a path between them so neither displacement nor distance are well defined for the jump. Teleportation is science fiction, but discontinuous curves can certainly be defined in maths. For example ##y=1/x## is discontinuous at the origin. You can't approximate that jump by a short straight line so the whole argument in my previous paragraph falls apart.Can you please explain a little bit more about the pathological functions that you mentioned?

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Imagine a particle spiralling into the origin in smaller and smaller circles in finite time. The average velocity depends on the decreading radius of the circle. But the average speed depends in how many times it goes round the circle.Can you please explain a little bit more about the pathological functions that you mentioned? seems interesting, and I would like to learn more about it.

You should be able to construct it so the the final velocity is zero but the final speed is infinite.

Something like that.

Mathematically, there will need to be some constraints on the trajectory so that the limits exist and are equal.

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Imagine what happens as the duration of those intervals of time approach zero.... I have understood the distinction between the magnitude of average velocity over an interval of time and the average speed over the same interval. The average speed it is defined as the total distance divided by time and not as magnitude of average velocity.

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As far as I know instantaneous speed is equal to the magnitude of instantaneous velocity

The other inequality is a consequence of the general inequality $$|\int \vec{v} dt|\leq \int|\vec{v}|dt$$ (which holds for any vector ##\vec{v}## and is a consequence of the

(In the case vector ##\vec{v}## is the acceleration it tell us that the magnitude of the average acceleration is less or equal to the average of the magnitude of acceleration.)

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