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If the energy of the light matches the threshold energy, then electrons will be emitted from the surface. If the energy of light is greater than the threshold energy, then the electrons emitted will have increasing kinetic energy.
Can something still be emitted with zero KE? It seems as though that would cause it to not move at all, and therefore not break free of the metal surface. However, it's possible I'm ascribing expectations of regular matter to the quantum level where they don't really apply.
I believe that saying there is 0 KE means that after the electron has been ejected, there is no excess energy leftover that would be considered Kinetic Energy. When the energies match up, all of it is used to eject the electron so there will be 0 KE.
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