I Instantaneous velocity - displacement and distance

[adjurovich]

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?​

 

[Orodruin]

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?

Why do these two different quantities result in the same thing?​

They do not. Not generally.

 

[adjurovich]

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?

 

[Orodruin]

The (signed) distance from the origin is the position in one dimension.