Q$3.$

$($a$)$ An $8cm$ diameter fire hose is maintained at a gauge pressure of $P_0$ by a fire engine for pumping water. There is a nozzle at the end of the hose, reducing the diameter down to $2cm.$ The horizontal water jet, exiting the nozzle end, impinges on a vertical wall. Draw a fully labelled diagram of the hose, nozzle section and vertical wall. Include a control volume enabling a momentum balance for the water jet and the vertical wall.

$($b$)$ Assume that the average velocity of the water in the hose is $u_0=0\mathrm{.}5m{s}^{-1}.$ Calculate the average velocity of the water at the nozzle exit, $u_1.$

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$($c$)$ Calculate

$($i$)$ the water volume flow rate within the hose;

$($ii$)$ the gauge pressure in the hose.

If needed, assume a value of the water velocity, $u_1,$ for part $($b$).$ State all the assumptions made to derive your result.

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$($d$)$ Apply the momentum balance within the water jet$/$wall region to calculate both intensity and direction of the reaction force exerted by the stationary vertical wall on the water jet. Start your analysis with the usual momentum equation and explain why you ignore any of its terms in this analysis.

$($e$)$ Apply the momentum balance within the water jet$/$wall region to calculate both intensity and direction of the reaction force exerted by a stationary wall on the water jet when the wall is tilted towards the nozzle by $30$ with respect to the vertical. Start your analysis with the usual momentum equation and explain why you ignore any of its terms in this analysis. State any assumptions made. For full marks, draw a fully labelled diagram of the water jet and tilted wall, including the control volume used for the calculation of the force.