Is there any standard terminology for the derivative of the magnitude of velocity with respect to time (suitable for use in first-year Calculus)? The word ‘acceleration’, in its technical sense, is exactly what I am not looking for; it is the derivative of the velocity itself, but I want the derivative of its magnitude, the speed.
This is a useful concept, because it matches the colloquial sense of the word ‘acceleration’; if this quantity is positive, then an object is speeding up (accelerating), if it is negative, then it is slowing down (decelerating), and if it is zero, then it's doing neither (although it might be changing direction). As $ v $ is sometimes used for the magnitude of the velocity vector $ \boldsymbol v $, so $ a $ is sometimes used for the quantity that I'm looking for, but notice that $ a = \mathrm d v / \mathrm d t $ is not the magnitude of the acceleration vector $ \boldsymbol a = \mathrm d \boldsymbol v / \mathrm d t $. (After all, $ a $ can be negative.)
In the $1$-dimensional case, $ \boldsymbol a = \pm a $ (although you might not want to use those symbols in this case), but the $ \pm $ is given by the sign of $ \boldsymbol v $ rather than by the sign of $ \boldsymbol a $. In more dimensions, you can write $ \boldsymbol a = a \boldsymbol T + \kappa v ^ 2 \boldsymbol N = a \boldsymbol T + \omega v \boldsymbol N $, where $ \boldsymbol T = \boldsymbol v / v $ is a unit vector in the direction of motion (so the $ a \boldsymbol T $ term is analogous to the $ \pm a $ in $ 1 $ dimension), $ \boldsymbol N $ is a unit vector in the direction of curvature, $ \kappa $ is amount of curvature, and $ \omega = \kappa v $ is angular speed. (In $3$ dimensions, angular velocity relative to the centre of the osculating circle is $ \boldsymbol \omega = \omega \boldsymbol B $, where $ \boldsymbol B = \boldsymbol T \times \boldsymbol N $ is the unit binormal vector, but angular speed makes sense in any number of dimensions.) So this is certainly a useful concept for analysing acceleration, in particular breaking acceleration down into change of speed and change of direction.
I have called this ‘colloquial acceleration’, but I haven't seen that used anywhere else. I've also called it ‘scalar acceleration’, and so have many other people, but that's not a good term if you're working in only $ 1 $ dimension (where everything is a scalar). I've also seen ‘tangential component of [the] acceleration’ (with ‘normal component of [the] acceleration’ for $ \kappa v ^ 2 = \omega v $), which is not so much a term for it as a definition of it, but it also doesn't work very well in $ 1 $ dimension.
So, is there a standard term for this? If people think that either of the vector-related terms is sufficiently standard that I could say with a straight face to my one-variable Calculus students ‹The term for the derivative of speed with respect to time is is ‘scalar acceleration’; next year, you will learn what ‘scalar’ means, but for now it is just technical jargon.›, then I will use it. Or if there is another standard term for it that I haven't heard of, then I will use that. If not, then I'll stick to ‘colloquial acceleration’, even though it is clearly not standard, since it's easy to explain.
