(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

(i) `|(dV)/(dt)|` is the magnitude of total acceleration. While `(d|V|)/(dt)` represents the time rate of change of speed (called the tangential acceleration, a component of total acceleration) as `|V|=v`.
(ii) These two are equal in case of one dimensional motion (without change in direction).
(iii) In case of uniform circular motion speed remains constant while velocity changes.
Hence, `(d|V|)/(dt)=0` while `|(dv)/(dt)|ne0`.
(iv) `(d|V|)/(dt)ne 0` implies that speed of particle is not constant. Velocity cannot remain constant if speed is changing. Hence, `|(dV)/(dt)|` cannot be zero in this case. So, it is not possible to be have `|(dv)/(dt)|=0` while `(d|V|)/(dt)ne 0`