To understand mass flow problems, first consider an object of mass m and velocity v that ejects a small sub-element. Let the small element be mass Δmf with a velocity vf relative to the object. The the object's new velocity will be v + Δv.
If a system containing the object and the element of mass is chosen such that no external forces act on it, then the total linear momentum must be conserved before and after the emission, giving
This simplifies to
- Δmf Δv
+ mf Δvf = 0
The higher order term, Δmf Δv, will be very smaller when compared to the other terms due to the double delta. This term can be assumed to be zero.
Now, assume that the object emits a continuous flow of mass Δmf
during the period Δt. Divide by Δt, and take the limit as Δt approaches 0,