Consider the spillway shown below. During normal operating conditions the spillway can handle a maximum volume flow rate of $50\frac{m^3}{s}$ before the water overflows the weir. In storm situations where the water passes over the weir, a porous bed was installed to absorbed the overflow water and keep the outlying area from being flooded. The water absorbs into the porous gravel at a uniform vertical velocity of $10m\frac{m}{s}.$

For a storm where the volume flowrate of water increases to $100\frac{m^3}{s},$ the height of the flow upstream of the weir is $h_1=2m,$ and the height of the flow downstream of the weir is $h_2=0\mathrm{.}5m$; if the width of the spillway is $5m,$

Determine:

$($a$)$ the length of the porous bed, $L_{\text{g}\text{r}\text{a}\text{v}\text{e}\text{l}},$ required to completely absorb the storm water.

$($b$)$ the force exerted by the river on the weir.

All Control Volumes must be drawn for full credit.

Assume:

The density of water is $1000k\frac{g}{m^3}.$

Hydrostatic pressure upstream and downstream of the weir.

Atmospheric pressure is negligible.

Friction at the bottom is negligible.

Problem $1.$