Parametric speed

$\sqrt{[x'(t)]^2+[y'(t)]^2}$ — the rate of distance traveled along the curve, the arc-length integrand.

A particle's speed along a parametric curve is $\sqrt{[x'(t)]^2+[y'(t)]^2}$ — the magnitude of its velocity vector $\langle x'(t),y'(t)\rangle$ . It's exactly the integrand of the arc-length formula: integrating speed over time gives total distance traveled, and speed equal to $0$ at some instant means the particle momentarily stops (a potential cusp in the curve).

Examples

For $x=t^2$ , $y=t^3$ : speed $=\sqrt{(2t)^2+(3t^2)^2}=\sqrt{4t^2+9t^4}$ , which is $0$ exactly at $t=0$ .