$1.)$ The pipe bend shown below discharges water into the atmosphere at section $2($where $\backslash (\backslash$mathrm$\{$D$\}\_\{2\}\backslash )$ is defined$).$ The bend lies in a horizontal plane $($i$.$e$.,$ elevation is not changing, and the diagram below provides a plan view$).$ Assume friction and viscous forces are negligible. The volume of water in the bend between sections $1$ and $2$ is $0\mathrm{.}25$ cubic meters at any point in time, while the mass of the pipe bend itself $($i$.$e$.,$ the pipe material$)$ is $250$ kg $.$ The diameter at section $1$ is $60$ cm $,$ the diameter at section $2$ is $10$ cm $,$ and the velocity of the water exiting section $2$ to the atmosphere is $\backslash (15\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ By the end of this problem, we will determine the force components $($in the $\backslash ($ x$,$ y $\backslash ),$ and $\backslash ($ z $\backslash )$ directions$)$ at the flange required to hold the bend in place.

a$)$ Find the velocity of water entering the bend through section $1,$ where the diameter is $60$ cm $.(5$ points$)$

b$)$ Find the volumetric flow rate of water passing through the bend. $(5$ points$)$

c$)$ Find the water pressure acting on section $1,$ assuming viscous forces are negligible. $(5$ points$)$

d$)$ Apply the momentum equation in the x$-$direction $($the horizontal direction of flow entering section $1)$ to calculate the x$-$component of the force at the flange. Hint: Be careful of the sign on the momentum term at section $2.$ It is flowing out of the bend, which would typically mean it is positive, but its momentum is flowing opposite the positive direction of the $\backslash ($ x $\backslash )-$axis defined in the diagram... $(5$ points$)$

e$)$ Apply the momentum equation in the $\backslash ($ y $\backslash )-$direction $($the horizontal direction perpendicular to flow entering section $1)$ to calculate the y$-$component of the force at the flange. Hint: Be careful of the signs on the momentum term$($s$)$ again. $(5$ points$)$

f$)$ Apply the momentum equation in the z$-$direction to calculate the z$-$component of the force at the flange. $(5$ points$)$