A thorium nucleus of mass M=3.85×10−25kg is traveling in the +x-direction with vi=600m/s . The thorium nucleus then undergoes gamma decay, in which a gamma ray photon is emitted with wavelength λγ=0.004nm , in the +x-direction as shown in the figure. Which of the following gives the best estimate for the final velocity vf of the thorium nucleus after the photon is emitted?

you'd use the equation that establishes a relationship between  energy  and  wavelength, which looks like this

e=h⋅cλ, where

λ  - the wavelength of the photon; c  - the speed of light, usually given as  3⋅108m/s; h  -  planck's constant, equal to  6.626⋅10−34j s

plug the value you were given for the energy of the photon in the above equation to get the wavelength

e=h⋅cλ⇒λ=h⋅ce

λ=6.626⋅10−34js⋅3⋅108ms−1.257⋅10−24j=1.581⋅10−1m

the answer should be in the picture.