I'm working on a problem to derive E=mc^2 using conservation of momentum and center of mass. We have a hollow block of length L and mass M. A photon passes through taking mass m and adding it to the other side at speed C. Here's what I've got:
The momentum p of the photon si the same as that of the box. So we have that E/C, which is the momentum of the photon=$m_{box}v_{box}$. The time it takes would be L/C and hence the distance would be vL/C. Let's say that the box moved a distance of x after the photon came in. What is that distance x?
Here's what I have done so far:
-The system is the box and the mass m being transported by the photon from one end to the other.
My reference point I will set is the edge of the box where the mass m is before displacement. Therefore,
$$X_{CoM}=1/M(LM/2)=L/2$$
The center of mass is L/2 from the reference point.
To find the distance x, I set L/2 as the following:
$$L/2=1/M(M(L/2-x)+m(L-x))$$
I think there's something wrong with this equation because at the end I'm getting that x=mL/M+m. Can someone help me out here? Thanks in advance