The "refresh rate" of the human eye/brain for whole pictures is about 15-20 times per second. So, in theory, you could archive smoothness with that frame rate. In reality you will obviously don't know the exact moment when the brain refreshes a image. So, to guarantee a new picture for the brain every time it refreshes, you have to have a much higher frame rate. About 30 FPS is enough.
But the eye or brain doesn't stop absorbing visual information between refreshes, everything in between is also processed at a much lower level of processing effort. Especially fast movements and overall information density are still recognized. That enables you to see improvements in picture quality up to 120 FPS and beyond. But that also makes it hard to put the finger on what it is this additional FPS do, they never reach your consciousness the way the initial frames do.
I have seen a great video comparison between 30, 60 and 120 FPS once, I'm looking for it now.