Calculating Reflection Angle and Frequency for Moving Mirror?

[xayon]

Homework Statement

A mirror is moving uniformly in a direction normal to its plane with velocity v=βc. Given the angle of incidence and frequency of an incident photon (θ_e,nu_e in the figure), calculate the reflection angle and the observed new frequency (θ_i,nu_i in the figure).

Also prove that:
sin(θ_e)/(1+βcos(θ_e)) = sin(θ_i)/(1-βcos(θ_i))

Homework Equations

It has been suggested in a couple of threads to use Lorentz transformations, but i seem to have problems dealing with them.

The Attempt at a Solution

As said, my attempts to use them have been an utter failure. In the mirror's system the angles should be the same, but i get stuck at this point.

 

[tiny-tim]

welcome to pf!

hi xayon! welcome to pf!

It has been suggested in a couple of threads to use Lorentz transformations, but i seem to have problems dealing with them.

yes, the Lorentz transformation should do it …

show us how far you've got, and where you're stuck ​

 

[haruspex]

Does it help to think about the momenta in the X and Y directions?

 

[xayon]

Yes, I've thought of the momentum, h*nu. Is the frequency of the incident photon as measured in the mirror frame the same as the one from an external observer?
The other thing to take into account is that in the mirror frame the angles are the same isn't it? But again, by Doppler effect, the frequency of the photon would be different...

 

[haruspex]

Yes, I've thought of the momentum, h*nu.
[/QUOTE}
Ok, but have you thought about the x and y components of momentum? The ratio of these should give the tan of the angle. The answer will be different in the two frames.

Is the frequency of the incident photon as measured in the mirror frame the same as the one from an external observer?

No.

The other thing to take into account is that in the mirror frame the angles are the same isn't it? But again, by Doppler effect, the frequency of the photon would be different...

In the mirror frame, there'll be no change in frequency.

 

[xayon]

Ok, having Lorentz transformations for the momentum:
$E'=\gamma(E-\beta cp_{x})$
$cp'_{x} =\gamma(cp_{x}-\beta E)$
$cp'_y=cp_y$
$cp'_z=cp_z$

Where the primed frame is the mirror's one.

As haruspex pointed, I have divided the x and y components of the momentum, getting two expressions for $tan(\theta _{e})$ proving the statement. From that i can get $\theta_{i}$ in terms of $\theta _{e}$ and v.

Finally getting $\nu$ from the 1st transformation.

Am I right?