Question #42069

An atom of mass $m$ is injected into a gas medium at $t=0$ with an initial velocity $v_0 \hat{i}$ relative to the lab frame. The gas medium flows with a uniform steady velocity $u_0 \hat{j}$ relative to the lab frame. The gas exerts a viscous drag force $\vec{F} = -b \vec{v}_{\text{rel}}$ on the atom, where $b$ is a positive constant and $\vec{v}_{\text{rel}}$ is the velocity of the atom relative to the gas. Exactly at $t=0$, the atom emits a photon of energy $E$ in the $+x$ direction. Assuming the atom does not emit any further photons, what is the total displacement vector of the atom relative to the gas medium from $t=0$ to $t \to \infty$? (Here $c$ is the speed of light in vacuum).