Proving that acceleration perpendicular to velocity only changes it's direction [duplicate]

Question

In a recent class, I learned about centripetal acceleration and that if a body moves in uniform circular motion the direction of velocity continuously changes implying presence of an acceleration. My professor derived the formula for it and also showed that it's always perpendicular to velocity.

However we were told that converse is also true which left me a little confused. how can we prove converse that if acceleration is perpendicular to velocity it will only change direction?

I tried from the fact that if acceleration is perpendicular to velocity it will have no component along it therefore will not be able to change the magnitude of velocity but since it will be acting it will have to bring some change and therefore will change direction of velocity.

is there any better way to prove this?

Answer 1

I'm not sure it's better, but you can write the time rate of change of the magnitude of the velocity squared as \begin{equation} \frac{d}{dt} v^2 = \frac{d}{dt} \vec v \cdot \vec v = 2 \dot {\vec v} \cdot \vec v =2\vec a \cdot \vec v \,. \end{equation} If $\vec a$ is perpendicular to $\vec v$ the change in $v^{2}$ (and thus the change in $|\vec{v}|=\sqrt{v^{2}}$) is zero.