but why is ##\dot m## effectively constant over the interval ξ? I never said that ξ is an infinitesimal time interval, it could be for example 5 seconds?
Yes sure.
ξ is a constant in the equation and is an arbitrary time step. Think of it as Δt. (Δt > 0)
After a certain time ξ, the momentum is the rocket times its velocity and the mass times its velocity, which is ejected downward.
At the time t + ξ, however, one must take into account that...
My reasoning was this:
After a certain time ξ, the momentum is the rocket times its velocity and the mass times its velocity, which is ejected downward.
At the time t + ξ, however, one must take into account that mass which was ejected at the time t has a higher velocity than particles which...