I Instantaneous velocity - displacement and distance

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Instantaneous velocity is defined as the first derivative of displacement with respect to time and can also be expressed as the first derivative of distance with respect to time. While both definitions yield the same result in one-dimensional motion, they do not generally equate in multidimensional scenarios. The discussion highlights the importance of context, particularly in one-dimensional cases where signed distance from the origin represents position. Clarifying these distinctions is crucial for understanding the nuances of motion in physics.
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.
 
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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