case 1 is simple as the discharge is given and all I have to do is use some formulas. But case number 2 is different as discharge is not given. How do I go about to solve this?
So the general approach is to use the bernoulli equation and the continuity equation for open channel flow. Here we have 2 unknowns, the discharge $Q$ and $d_2 = d_c$, the critical depth, which is assumed to occur on the crest of the weir. $v_2 = v_c$ for critical flow. I need to solve for $d_2$ in terms of the unknown $Q$. How do I do this with iteration and not solve with a cubic equation?