# How can speed be the magnitude of velocity

Say speed is the magnitude of velocity then: \begin{align*} v &=\left|\vec{v}\right|\\ \frac{D}{\Delta t} &= \left|\frac{\Delta \vec{x}}{\Delta t}\right|\\ \end{align*} since $$\Delta t$$ is always positive, we get: $$D = \left|\Delta \vec{x}\right|$$ which is not true.

$$D$$ = distance travelled

$$\Delta \vec{x}$$ = displacement

$$||$$ is magnitude

$$\Delta t$$ is time taken

$$v$$ is speed

$$\vec{v}$$ is velocity

Let me be specific here. Speed is not the magnitude of velocity but instantaneous speed is the magnitude of instantaneous velocity. As pointed out by Brain Stroke Patient, this happens in the limiting case as $$lim \Delta t \rightarrow 0$$. In that case, $$|\Delta\vec{x}|\rightarrow D$$.
$$\vec{v} = \Delta \vec{x}/\Delta t$$ for infinitesimally small displacement and time. In that limit D is equal to $$\Delta x$$. If the displacement isn't infinitesimal then you have the average velocity and that need not be equal to, in magnitude, to the average speed.