$3-$ In a rectangular channel with a width of $5$ m $,$ the flow rate is $20\frac{m^3}{s}.$ A sluice gate has an opening of $0\mathrm{.}67$ m $.$ The water depth in the vena$-$contracta region is $a=0\mathrm{.}4m,$ and the downstream water depth $(h_2)$ is $2\mathrm{.}5$ m $.$ Solve the following:

a$)$ Prove that a hydraulic jump will occur downstream of the gate.

b$)$ Calculate the energy loss in the hydraulic jump and determine the power dissipated by the jump.

c$)$ Considering that some energy is lost while passing beneath the sluice gate is equal to $5\%$ of the velocity head, $,$ where $V$ is the velocity in the contracted region$),$ find the water depth $h_0$ before the gate.

d$)$ Determine the force exerted on the gate, Fg$.$

e$)$ If the downstream water depth $h_2$ is controlled at $3$ m instead of $2\mathrm{.}5$ m $,$ analyze the flow conditions downstream of the gate with this depth. $($Hint: After identifying the type of flow, you can apply the momentum and continuity equations to derive an expression for the downstream depth. Use this equation to calculate the depth just after the gate.$)$