This is a stupid question but what is the proper name of this formula? Been trying to look for a proper definition for this but can't find it online. It's been a long time since I had to use these that my brain is drawing a blank.
$$||a||=\sqrt{\left(\frac{{\rm d}^{2}y}{{\rm d}t^{2}}\right)^{2}+\left(\frac{{\rm d}^{2}x}{{\rm d}t^{2}}\right)^{2}}$$
Edit: just found it used in this question: Acceleration of a particle described by parametric equations
So it's the magnitude of the vector's speed?
If $\mathbf{c}(t) = (x(t),y(t))$, then $\frac{ds}{dt}$ is the derivative of arc length with respect to $t$. If you think of $\mathbf{c}(t)$ as the position of a particle, $\mathbf{c}'(t)=(x'(t),y'(t))$ is its velocity and $||\mathbf{c}'(t)|| = \sqrt{x'(t)^2 + y'(t)^2}$ is its speed.