What does `d|vec(v)|//dt` and `|d vec(v)//dt|` represent ? Can these be equal ? Can. (a) `d|vec(v)|//dt = 0` while `|d vec(v)//dt| != 0` (b) `d|vec(v)|//dt != 0` while `|d vec(v)//dt| = 0`?

Explanation

`d |vec(v)|//dt` represents time rate of change of speed as `|vec(v)| = v`, while `|d vec(v)//dt|` represents magnitude of acceleration. If the motion of a particle is accelerated translatroy (without change in direction)
as `vec(v) = |vec(v)| hat(n), (d vec(v))/(dt) = (d)/(dt) [|vec(v)| hat(n)]`
or `(d vec(v))/(dt) = hat(n) (d)/(dt) |vec(v)|` [as `hat(n)` constant]
or `|(d vec(v))/(dt)| = (d)/(dt) |vec(v)| (!= 0)`
However, if the motion is uniform translartory, both these will still be equal but zero .
(a) The given condition implies that:
`|d vec(v) //dt| !=`, i.e., |acc.| `!=0` while `d|vec(v)|//dt = 0`,
i.e., speed = constant
This actually is the case of uniform circular motion. In case of uniform circular motion
`|(d vec(v))/(dt)| = |vec(a)| = (v^(2))/(r) =` constt. `!= 0`
while `|vec(v)| =` constt. i.e., `(d)/(dt) |vec(v)| = 0`
(b) `|d vec(v)//dt| = 0` means `|vec(a)| = 0` means `|vec(a)| = 0`, i.e., `vec(a) = 0`
or `(d vec(v)//dt) = 0` or `vec(v)=` constt.
And when velocity `vec(v)` is constant speed will be constt.,
i.e., speed `= |vec(v)| =` constt. or `(d)/(dt) |vec(v)| = 0`
So, it is not possible to have `|(d vec(v))/(dt)| = 0` while `(d)/(dt) |vec(v)| != 0`