Consider an excited hydrogen atom in state `n` moving with a velocity `v(v lt lt c)`. It emits a photon in the direction of its motion and changes its state to a lower state `m`. Apply momentum and energy conservation principle to calculate the frequency v of the emitted radiation, compare this with the frequency `v_(0)` emitted if the atom were at rest.

Explanation

By momentum conservation
`mv=mv_(1)+(hv)/(c )`
`v_(1)=v-(hv)/(mc)`
By energy conservation
`(1)/(2) mv^(2)+DeltaE=(1)/(2) mv_(1)^(2)+ hv`
" " `=(1)/(2)m(v-(hv)/(mc))^(2)+hv`
" " `=(1)/(2)mv^(2)-(1)/(2)m.2v(hv)/(mc)+(1)/(2)m(h^(2)v^(2))/(mc)+ hv`
`Delta E= hv(1-(v)/(c )) { because (h^(2)v^(2))/(m^(2)c^(2)) "is negligible"}`
When atom is at rest
`0=-mv_(2)+(hv_(0))/(c )`
`v_(2)=(hv_(0))/(mc)`
`DeltaE=(1)/(2) mv_(2)^(2)+hv_(0)`
`=(1)/(2)m(h^(2)v_(0)^(2))/(m^(2)c^(2)+hv_(0)`
`hv_(0) [ because (h^(2)v_(0)^(2))/(m^(2)c^(2)) " is negligible"]`
`Delta E=hv_(0)=hv(1-v//c)`
`v=(v_(0))/((1-v//c))=v_(0)(1-v//c)^(-1)=v_(0)(1+(v)/(c))`