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What is the conceptually difference between the two: $$\frac{d|\vec{r}|}{dt}=\frac{\vec{r}\cdot\frac{d\vec{r}}{dt}}{|\vec{r}|}\neq|\dot{\vec{r}}|\equiv \bigg|\frac{d\vec{r}}{dt}\bigg|$$ Unidimensionally it appears an equality, but Im having trouble to think qualitatively.

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  • $\begingroup$ Does $|\dot{\vec{r}}|$ mean the absolute value of velocity, i.e. $|\frac{d\vec{r}}{dt}|$? $\endgroup$
    – RC_23
    Jun 3 at 16:35

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The first describes the rate at which the distance between the object and the (often arbitrary) origin is changing, whereas the second is the actual speed of the object (the speed being the magnitude of the velocity vector, which is the derivative of the position vector).

These two quantities are clearly different. For an object moving at constant speed $v$ in a circle, the former is zero, whereas the latter is exactly $v$.

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