(a) What does `|(dv)/(dt)|` and `(d|v|)/(dt)` represent? (b) Can these be equal?

Explanation

`|vecv|//dt` represents time rate of change of speed as `|vecv|=u`, while `|(d vec v)/(dt)|` represetns magnitude of acceleration.
If the motion of a particle has translational accelerationn (without change in direction)
as `vecv=|vecv|hatn,(dvecv)/(dt)=(d)/(dt)[|vecv|hatn]`
or `(dvecv)/(dt)=hatn(d)/(dt)|vecv|` [as `hatn` is constant]
or `|(dvecv)/(dt)|=(d)/(dt)|vecv|(ne0)`
In this case both these will be equal and not equal to zero.
(a). The given condition implies that:
`|dvec(v)//dt|ne 0`, i.e., `|ac c.|ne0` while `d|vecv|//dt=0`
i.e., speed `=` constant
This actually is the case of uniform circular motion. in case of uniform circular motion
`|(dvecv)/(dt)|=|veca|=(u^(2))/(r)=` constant
While `|vecv|=` constant i.e., `(d)/(dt)|vecv|=0`
(b) `|dvec v//dt|=0` means `|veca|=0`, i.e., `vec a=vec 0`
or `((dvecv)/(dt))=vec0 or vecv=` constant
And when velocity `vecv` is constant speed will be constant.
i.e., speed `=|vecv|=` constant or `(d)/(dt)|vecv|=0`
So, it is not possible to have `|(dvecv)/(dt)|=0`
While `(d)/(dt)|vecv|ne0`.