Guys the below link is a question already asked in this website and I have referred the answers given by the users and some of them were good ,but I have few doubts and I wanted to ask it in a comment but my reputation wasn't 50,so i'm posting it as a question. Sorry for the duplication! the link is this: Why isn't momentum conserved in this pulley problem?. My doubt is this: In the solution give in the users question the impulse imparted to the partice should be mV-mv right? What I mean is that it should be final momentum -initial momentum right? And,impulse imparted for block should be -mV right? i.e it should negative because the block moves up. Please forgive my bad presentation of the problem.
The outlined solution has up positive, but $v$ and $V$ are implicitly positive in a downward direction. With up, $v$, and $V$ positive, the ball's initial momentum is $-mv$ and it's final momentum is $-mV$. The change in momentum is thus $(-mV) - (-mv)$ or $mv-mV$.
Note that with up positive, it would perhaps have been better to write equation (ii) as $$\int(T-N)\,dt = -mV$$ Negating both sides results in equation (ii) as written in the outlined solution.