This problem involves an accelerating object $($a rocket$)$ whose mass is changing while a force acts on it$.F=\frac{dp}{dt}=\frac{d}{dt}(mv)=m(\frac{dv}{dt})+v(\frac{dm}{dt})$ and we can't ignore $\frac{dm}{dt}$ in this kind of situation.

The "rocket equation" is a very simple way to calculate the change in velocity given its initial velocity, the relative velocity that the propellant is exhausted $(v_e)$ and the initial and final masses and $($:$m_f$