Question #43697

A beam of unstable particles, each of mass $m$ (in $\text{kg}$) and initial velocity $v_0$ (in $\text{m s}^{-1}$), enters a gaseous medium at $t=0$. The particles undergo radioactive decay with a decay constant $\lambda$ (in $\text{s}^{-1}$). The medium has a uniform number density $n$ (in $\text{m}^{-3}$) and the interaction cross-section between the particles and the medium molecules is $\sigma$ (in $\text{m}^2$). As the particles traverse the medium, they experience a viscous drag force $F = -bv$, where $v$ is the instantaneous velocity and $b$ (in $\text{kg s}^{-1}$) is a positive constant. Given that the decay constant is exactly equal to the drag coefficient per unit mass ($\lambda = b/m$), what is the total probability $P$ that a particle will undergo a collision with a medium molecule before it decays? Let the dimensionless parameter $\beta = \frac{n \sigma m v_0}{b}$.