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In classic mechanics when 2 bodies have an elastic collision we use momentum and energy conservation laws like described here

I want to consider system which consists of photon (1 or many identical) with frequency ${f_0}$ and 1 ideal mirror with macroscopic mass m, like 1 gramm or so and zero speed.

I assume when photon will be reflected from ideal mirror, it will be kind of elastic collision because there is no energy loss.

If I write momentum / Energy conservation laws like this

Photon momentum is $\Large \frac{h{f_0}}{c}$

Photon energy $\Large h{f_0}$

Momentum conservation law

$\Large\frac{h{f_0}}{c} = m v - \frac{h{f_1}}{c}$

Where ${f_1}$ new photon frequency after reflection.

$v$ - mirror speed after reflection

Minus sign before photon's momentum, because photon momentum change direction.

Energy conservation law

$\Large h{f_0} = \frac{mv^2}{2} + h{f_1}$

If I want to consider many photons, I just add N (number of photons) before every $hf$.

I was expecting to get for $f_1$ something like Doppler effect, but result is different.

My question is: Are these equations and assumptions correct or they are not applicable in this case?

UPDATE1:

Should I use relativistic equations for momentum and energy $\Large p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}, E=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$, if my mirror moves with the speed much less then speed of light because another body is photon? Of course I have to use them, if my mirror has relativistic speed.

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