The arc-length parametrisation of curve has [[Unit speed parametrisation|unit speed]]. *Proof sketch.* Using chain rule $\frac{d\underline{r}}{ds}= \frac{dt}{ds} \frac{d\underline{r}}{dt} \implies \left\lvert \left\lvert \frac{d\underline{r}}{ds} \right\rvert \right\rvert = \left\lvert \frac{dt}{ds} \right\rvert \; \left\lvert \left\lvert \frac{d\underline{r}}{dt} \right\rvert \right\rvert $ Note that $\left\lvert \left\lvert \frac{d\underline{r}}{dt} \right\rvert \right\rvert = \frac{ds}{dt}$ so we're done.