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I heard some one talked about the instantaneous and average velocities.

He was using $\Delta t$ to denote a time frame, $dt$ denote a time point.

average velocities $\bar{v} = \dfrac{\Delta s}{\Delta t}$

the $\Delta t$ part is indeed common. my concern is about the $dt$ part

wiki use the notation

instantaneous velocities $v = \dfrac{ds}{dt}$

Is it reasonable and common to interpret this way?

whnlp
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2 Answers2

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$\text dt$ is not a point in time. It is an infinitesimal time interval. Physically, you could think of it as a time interval that is much much smaller than the relevant time scale of the system. Mathematically, it is the limit of $\Delta t$ as it approaches $0$ (not equal to $0$).

This just comes from the definition of the limit: $$v=\lim_{\Delta t\to0}\frac{x(t+\Delta t)-x(t)}{\Delta t}$$

Limits are not the same thing as equality. Plugging in $\Delta t=0$ makes the above definition undefined.

BioPhysicist
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dt is indeed an infinitesimally small amount of time. It is so small, infact, that no matter in what directions and manner a body is undergoing motion, it's motion is always straight line for time dt.