) Show that if the acceleration of an object is always perpendicular to the velocity, then the speed is of the object is constant.

Question

Since acceleration is the derivative of velocity, it should be tangent to it. How do I show this mathematically?

Answer 1

$$0=\vec{a}.\vec{v}=\vec{\dot{v}}.\vec{v}=\frac12(|\vec{v}|^2)'$$ then $|\vec{v}|=C$.