Reservoirs $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ are connected by a pipe with a sudden expansion. Each of the two pipe segments is $10$ m long. Friction head loss in the pipes is given by $\backslash (0\mathrm{.}02\backslash$frac$\{$L$\}\{$D$\}\backslash$frac$\{$V$^\{2\}\}\{2$ g$\}\backslash ),$ where $\backslash ($ L $\backslash )$ is the pipe length, $\backslash ($ D $\backslash )$ is the pipe diameter, and $\backslash ($ V $\backslash )$ is the average velocity in that pipe. The entrance from Tank $\backslash ($ A $\backslash )$ to Pipe $1$ is smooth. Accounting for total head loss $($including transitions$),$ calculate

$($a$)$ the velocities in the two pipes, $\backslash ($ V$\_\{1\}\backslash )$ and $\backslash ($ V$\_\{2\}\backslash ),$

$($b$)$ the fluid discharge, $\backslash ($ Q $\backslash ),$ between the tanks,

$($c$)$ the contribution of each term to total head loss: friction in Pipe $1,$ friction in Pipe $2,$ sudden expansion between Pipes $1$ and $2,$ and expansion into Tank B$.$ Check that all the head loss components sum to the correct total. Express each of the four contributions as a fraction.

Assume steady flow.