$---$

$**$Question:$**$

Given the following parameters for a water rocket:

$-$ Water volume added to the bottle: $\backslash ($V$\_\{$h$2$o$\}=0\mathrm{.}04\backslash ,\backslash$text$\{$l$\}\backslash )$

$-$ Total empty volume of the bottle: $\backslash ($V $=1\backslash ,\backslash$text$\{$l$\}\backslash )$

$-$ Specific heat ratio: $\backslash (\backslash$gamma $=1\mathrm{.}4\backslash )$

$-$ Target horizontal distance: $\backslash ($Dis$\_$t $=200\backslash ,\backslash$text$\{$ft$\}\backslash )$

$-$ Diameter of the nozzle: $\backslash ($D$\_$n $=0\mathrm{.}009\backslash ,\backslash$text$\{$m$\}\backslash )$

$-$ Angle of the cone: $\backslash (\backslash$theta$\_\{$co$\}=70^\backslash$circ$\backslash )$

$-$ Diameter of the cone: $\backslash ($D$\_\{$co$\}=0\mathrm{.}077\backslash ,\backslash$text$\{$m$\}\backslash )$

$-$ Gravitational acceleration: $\backslash ($g $=9\mathrm{.}81\backslash ,\backslash$text$\{$m$/$s$\}^2\backslash )$

$-$ Mass of the rocket: $\backslash ($m $=1\backslash ,\backslash$text$\{$kg$\}\backslash )$

$-$ Atmospheric pressure: $\backslash ($p$\_\{$atm$\backslash \_$hg$\}=30\mathrm{.}21\backslash ,\backslash$text$\{$inHg$\}\backslash )$

$-$ Density of water: $\backslash (\backslash$rho$\_\{$h$2$o$\}=1000\backslash ,\backslash$text$\{$kg$/$m$\}^3\backslash )$

$-$ Density of air: $\backslash (\backslash$rho$\_\{$air$\}=1\mathrm{.}293\backslash ,\backslash$text$\{$kg$/$m$\}^3\backslash )$

$-$ Drag coefficient: $\backslash ($C$\_$d $=0\mathrm{.}92\backslash )$

Develop two equations to predict the following:

$1.**$Total pressure$**$ required in the rocket bottle to achieve the given horizontal target distance when the angle of launch $\backslash (\backslash$theta$\backslash )($Choose a reasonable number$)$ is known.

$2.**$Launch angle$**\backslash (\backslash$theta$\backslash )($Choose a reasonable number$)$ required for the rocket to reach the target distance when the total pressure in the rocket bottle is given.

Assume that the rocket$$s trajectory is influenced by both the pressure in the bottle and the angle at which it is launched. Use appropriate fluid dynamics, projectile motion, and aerodynamics principles to derive the equations based on the given parameters.

$---$