If we have a channel with varying width shown in Figure $2.$ The water depth and width at section $1$ and $2$ are $y_1(2m),b_1(5m)$ and $y_2(1\mathrm{.}5m)b_2(3m)$ respectively. The distance between these two sections is $10$ m $.$

$(1)$

If the channel is flat $(=0),$ and there is no energy loss $(h_L=0)$ during the change of water depth. Please determine Q$.$

$(2)$

If the channel has a slope with $=0\mathrm{.}57(tan()=0\mathrm{.}01),$ and there is a $0\mathrm{.}5$ m energy head loss $(h_L)$ during the change of water depth. Please determine Q$.(15\%)$

Hint: Use the continuity and energy equation.

Figure $2.$