1.

In a reference frame `K`a photon of frequency `omega` falls normally on a mirror approaching it with relativistic velocity `V`. Find the momentum inparted to the mirror during the reflection of the photon (a) in the reference frame fixed to the mirroe, (b) in the frame `K`.

Answer» (a) In the reference frame fixed to the mirror, the frequency of the photons is, by the Doppler shift formula
`overset_(omega) = omega sqrt((1+beta)/(1-beta)) (=omegasqrt(1-beta^(2))/(1-beta))`
In this frame momentum imparted to the mirror is
`(2cancelh overset_(omega))/(c) = (2cancelh omega)/(c) sqrt((1+beta)/(1-beta))`,
(b) In the `K` frame, the incident particle carries a momentum of `cancelh omega//c`and returns with momentum
`(cancelh omega)/(c)(1+beta)/(1-beta)`
The momentum inparted to the mirror, then, has the magnitude
`(cancelh omega)/(c)[(1+beta)/(1-beta)+1] = (2cancelh omega)/(c)(1)/(1-beta)`
Hence `beta = (V)/(c)`.


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