# Centripetal force and constant speed [duplicate]

This may seem like a trivial question, and I'm having a little trouble fleshing out my issues with the answers Ive seen so be prepared for some discussion on your answer. But why does centripetal force not change the speed?

I keep coming to this crux. If you have an initial velocity $v$ and then have a force $f$ at right angles to it. Would the new velocity not be greater than $v$ by pythagoras? How can it remain constant?