Effect of Radiation Pressure on a Moving Object

[wnvl2]

On a Dutch forum, there is a discussion about the following problem:

A $1 \mathrm{kg}$ mass slides across a frictionless floor with an initial velocity of $10\, \mathrm{m/s}$. The top surface of the mass has an area of $1\, \mathrm{m^2}$. The floor is perpendicularly irradiated with light of intensity $1\, \mathrm{W/m^2}$. How long does it take for the speed of the mass to decrease to $1\, \mathrm{m/s}$?

This can be seen as an interaction between the mass, the floor, and the incoming radiation.
Can I assume that the horizontal momentum of the mass remains constant when viewed from the frame of the floor?

More precisely: does the relativistic horizontal momentum of the mass remain constant from the initial state (1), throughout the interaction, until the final state (2)? That is,

$$
\gamma_1 m_1 v_1 = \gamma(t) m(t) v(t) = \gamma_2 m_2 v_2
$$

Is it valid to calculate the increase in mass via the energy absorbed from the incoming radiation, and then determine the change in velocity by assuming that the relativistic horizontal momentum remains constant?

ps No idea what is wrong with the LaTeX rendering.
[Mentors' note: You needed double hash delimiters not single dollar signs for the inline delimters. We've fixed this post for you, but next time check out our Latex help guide]

 

[Ibix]

Didn't we discuss this already?

Yes, the point is that the horizontal momentum is conserved and the mass is the only thing with horizontal momentum. As long as you're allowed to assume there's no reflection and the radiation is perpendicular to the floor in the floor rest frame, anyway. And that it doesn't lose any energy by radiation during whatever period it's absorbing radiation.