>> -- The scalar acceleration has to do with speed. Speed is the
>> forward component of velocity.
>
> I am accustomed to velocity having components, x and y or tangential
> and radial
OK.
> but forward?
Why not? If we can define the forward direction,
we can project off a component in that direction,
nicht wahr?
> Is that in the instantaneous direction of the vector?
Yes.
> Will that not be differentially different in the next
> differentially different time?
Yes, it will be different.
> My limited understanding has speed as the "magnitude" of the
> velocity.
It's provably synonymous.
For any vector v (velocity vector or otherwise) that
happens to point in the forward direction (i.e.
aligned with the velcotiy, as discussed above), we
can write
|v| = sqrt(v dot v)
= sqrt(|v|^2)
= (v dot v) / |v|
= v dot (v / |v|)
= v dot (unit vector in the fwd direction)
= forward component of v.
Vectors are tricky in polar coordinates, because the
"radial" direction changes from place to place.
Trying to calculate things in terms of |v| is similarly
tricky, indeed trickier, because the notion of "forward"
is so changeable.