Definitely, physics forum is a go to ! Especially when chatGPT doesn't correct my approach instead just gives solution. I expect Physics forum to help me identify what mistake in making so that I can correct my thought process.
Unfortunately the Examiner hasn't provided any solution, neither any standard textbooks have a similar problem as far as i know. so my only go to was ChatGPT. It gave the solution :
$$v(t) = \frac{\rho Au^{2}t}{M_{C}+\rho Aut}$$
in my solution there is ##M_{c}## which the the mass of container and ##M## which is shorthand for ##M(t)##. There could be a confusion between those two since the question mentions only ##M##. However now that i have mentioned my notations, i think it would be clear. There is time dependence...
Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system
$$M(t) = M_{C} + m(t)$$
$$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$
$$P_i = Mv + u \, dm$$
$$P_f = (M + dm)(v + dv)$$
$$\Delta P = M \, dv + (v - u) \, dm$$
$$F =...