Instantaneous velocity - displacement and distance

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Discussion Overview

The discussion revolves around the definitions and implications of instantaneous velocity, specifically examining the relationship between displacement and distance. Participants explore whether these definitions yield the same results in various contexts, including one-dimensional scenarios.

Discussion Character

  • Technical explanation, Debate/contested, Conceptual clarification

Main Points Raised

  • Some participants assert that instantaneous velocity is defined as the first derivative of displacement with respect to time, while others emphasize that it can also be defined as the first derivative of distance with respect to time.
  • One participant questions the validity of equating these two definitions, suggesting that they do not generally yield the same results outside of specific cases.
  • A later reply seeks clarification on how the definitions might apply in a one-dimensional context, indicating a need for further explanation.
  • Another participant notes that the signed distance from the origin can be considered the position in one dimension, potentially linking the concepts of distance and displacement.

Areas of Agreement / Disagreement

Participants express disagreement regarding whether the definitions of instantaneous velocity as derived from displacement and distance are equivalent in general contexts. The discussion remains unresolved with competing views presented.

Contextual Notes

Participants have not reached a consensus on the conditions under which the definitions of instantaneous velocity may align or differ, particularly in multi-dimensional versus one-dimensional settings.

adjurovich
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Instantaneous velocity is defined as the first derivative of displacement with respect to time:

##\vec{v} = \dfrac{d \vec{r}}{dt}##
However, instantaneous velocity is also defined as the first derivative of function of distance with respect to time:

##v = \dfrac{ds}{dt}##
Why do these two different quantities result in the same thing? We can certainly find the distance traveled between two points if we know the displacement function, why?​
 
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adjurovich said:
Instantaneous velocity is defined as the first derivative of displacement with respect to time:

##\vec{v} = \dfrac{d \vec{r}}{dt}##
However, instantaneous velocity is also defined as the first derivative of function of distance with respect to time:

##v = \dfrac{ds}{dt}##
Says who and where? Except as a special case in a one-dimensional setting?

adjurovich said:
Why do these two different quantities result in the same thing?​
They do not. Not generally.
 
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Orodruin said:
Says who and where? Except as a special case in a one-dimensional setting?


They do not. Not generally.
How would you explain it in “special” one dimensional case?
 
The (signed) distance from the origin is the position in one dimension.
 

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