(i) What does `|(dv)/(dt)|` and `(d|V|)/(dt)` represent ?
(ii) Can these be equal ?
(iii) Can `(d|V)/(dt)= 0` while `|(dV)/(dt)ne0` ?
(iv) Can `(d|V|)/(dt)ne 0` while `|(dv)/(dt)|=0` ?

Explanation

`|(d overset(vec)V)/(dt)|` is the magnitude of total acceleration.
While `(d|overset(vec)V|)/(dt)` is represents the time rate of change of speed i.e. magnitude of tangential acceleration.
These two are equal only in case of one dimensional motion without change in direaction.
In case of uniform circular motion speed remains constant while velocity changes.
Hence, `(d|overset(vec)v|)/(dt)=0` is possible while `|(doverset(vec)V)/(dt)|neO`
`|(d overset(vec)V)/(dt)|ne O` implies that speed of the particle is not constant. `|(doverset(vec)V)/(dt)|=O` means acceleration has zero magnitude. This is not possible.