Instantaneous velocity - displacement and distance

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SUMMARY

Instantaneous velocity is defined as the first derivative of displacement with respect to time, represented mathematically as v = d r/dt. It can also be expressed as the first derivative of distance with respect to time, v = ds/dt. While these definitions may appear interchangeable, they do not yield the same results in general cases, particularly outside of one-dimensional scenarios. The discussion emphasizes that the equivalence of these two definitions holds true only under specific conditions, such as in one-dimensional motion.

PREREQUISITES
  • Understanding of calculus, specifically derivatives
  • Familiarity with the concepts of displacement and distance
  • Knowledge of one-dimensional motion principles
  • Basic physics concepts related to velocity
NEXT STEPS
  • Study the implications of derivatives in multi-dimensional motion
  • Explore the differences between distance and displacement in physics
  • Learn about the applications of instantaneous velocity in real-world scenarios
  • Investigate the mathematical proofs of velocity definitions in various dimensions
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Students of physics, educators teaching kinematics, and anyone interested in the mathematical foundations of motion and velocity concepts.

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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