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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?

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If the force had a component in the direction of the velocity it would increase the speed. If it had a component in a direction opposite to the velocity it would decrease the speed. Because the centripetal force is perpendicular to the velocity it does neither.

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