I tried to use conservation of momentum p_f = p_i and then sub for mass of radia

Question

I tried to use conservation of momentum p_f = p_i and then sub for mass of radiation as E = mc^2 But it left the box NOT MOVING which the second part of the question implies that the box IS MOVING


Einstien showed that there is mass associated with electromagnetic radiation. Consider a box of length L and mass M resting on a frictionless surface. At the left wall of the box is a light source that emits radiation of energy E, which is absorbed momentum of magnitude of
p = E/c
find the recoil velocity of the box such that momentum mechanics. When the light is absorbed at the right wall of the box, the box stops, so the total momentum remains zero. If we neglect the very small velocity of the box, the time it takes for the radiation to travel across the box is delta t = L/c. Find the distance move by the box in this time, show that if the centre of mass of the system is to remain at the same place, the radiation must carry mass
m=E/c^2

Explanation / Answer

After the box absorbs the photon it will not be moving (to conserve momentum) E/c = M V to conserve momentum while the box is moving V = E / (M c) t = L/c time for photon to cross box x = V t = E L / (M c^2) Center of mass of box moves M x Center of mass of photon moves L - x 0 = M x - m (L - x) for the center of mass to remain stationary M * E L / (M c^2) = m (L - E L / (M c^2) E/c^2 = m (1 - E/(M c^2) = m - m E/ (M c^2) m = (1 + m/M) E/c^2 Now m/M is a very small quantity compared to 1 So m = E/c^2 Total momentum is zero at all times and the center of mass is stationary at all times.

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