[Math] Why normal vector and acceleration aren’t the same

Question

I know the derivative of position vector function $r(t)$ is velocity, and its unit vector is tangent vector, therefore, the derivative of tangent vector is the unit vector of acceleration and normal vector. Why isn't that the case?

Answer 1

If speed $(\|r'(t)\|)$ is constant, then the acceleration must be normal to the direction of travel.

But if speed is variable then acceleration can be broken into a component parallel to the direction of travel and a component that is perpendicular to the direction of travel.