Consider the following phenomenon.
Particles each traveling with speed $v_i$ metres per second enter a chamber at a rate of $r$ particles per second.
Upon entering the chamber, each particle begins to reduce its speed from $v_i$ to $v_f$. Each particle then exits the chamber.
If I look at an individual particle entering the chamber at time $0$, it is travelling $Ce^{-At}$ metres per second after $t$ seconds. The constants $C$ and $A$ are the same for every particle.
I let this process run for some time. Is there an expression $n(v,t)$ describing the number of particles in the chamber which are travelling at speed $v_i \leq v \leq v_f$ at time $t > 0$?