If \[{{\lambda }_{0}}\] and \[\lambda \] be the threshold wavelength and wavelength of incident light, the
velocity of photoelectron ejected from the metal surface is
     (a) \[\sqrt{\dfrac{2h}{m}({{\lambda }_{0}}-\lambda )}\]
     (b) \[\sqrt{\dfrac{2hc}{m}({{\lambda }_{0}}-\lambda )}\]
     (c) \[\sqrt{\dfrac{2h}{m}\left( \dfrac{{{\lambda }_{0}}-\lambda }{\lambda {{\lambda }_{0}}} \right)}\]
     (d) \[\sqrt{\dfrac{2h}{m}\left( \dfrac{1}{{{\lambda }_{0}}}-\dfrac{1}{\lambda } \right)}\]

Hint: Threshold wavelength is defined as the minimum wavelength of incident light required in order to eject the electrons from the metal surface. Once the threshold is crossed the metal surface exhibits the photoelectric effect.

Complete step by step solution:

We already know that,
\[hv=\phi +Kinetic\,Energy\] (i)
     where, v=incident frequency
          Planck’s constant (h) = \[6.62\times {{10}^{-34}}\,J\sec \]
          ϕ=threshold energy
We have been given in the question that
\[\phi =\dfrac{hc}{{{\lambda }_{0}}}\] (ii)
The equation of threshold energy is given as
\[hv=\dfrac{hc}{\lambda }\] (iii)
And for Kinetic energy is given as,
\[K.E.=\dfrac{1}{2}m{{V}_{\max }}^{2}\] (iv)
From equation (ii)(iii)(iv) and (i), we get,
\[\dfrac{hc}{\lambda }=\dfrac{hc}{{{\lambda }_{0}}}+\dfrac{1}{2}m{{V}_{\max }}^{2}\]
On rearranging,
\[\dfrac{hc}{\lambda }-\dfrac{hc}{{{\lambda }_{0}}}=\dfrac{1}{2}m{{V}_{\max }}^{2}\]
We will take hc as common,
\[hc\left( \dfrac{1}{\lambda }-\dfrac{1}{{{\lambda }_{0}}} \right)=\dfrac{1}{2}m{{V}_{\max }}^{2}\]
Again, on rearranging,
\[\dfrac{2hc}{m}\left( \dfrac{1}{\lambda }-\dfrac{1}{{{\lambda }_{0}}} \right)={{V}_{\max }}^{2}\]
Finally, we get,
\[{{V}_{\max }}^{2}=\sqrt{\dfrac{2h}{m}\left( \dfrac{{{\lambda }_{0}}-\lambda }{\lambda {{\lambda }_{0}}} \right)}\]
So, the correct option is (c).

Additional Information:

The photoelectric effect can be explained by assuming that radiation consists of particles of light called photons and each photon carries a quantum of energy. The energy of a photon depends solely on its frequency, which is the frequency of the radiation
The maximum kinetic energy of a photoelectron is the difference between the energy of the incident photon and the work function of the metal, where the work function is the binding energy of electrons to the metal surface. Each metal has its own characteristic work function.

Note: The name threshold wavelength differs from literature to literature. Don’t get confused between threshold wavelength, threshold frequency or work function. Essentially all have the same meaning.