$1.$ A manned spacecraft get impacted by space debris resulting in a hole as shown in the figure. Oxygen$-$rich air $($gas constant, $\backslash ($ R$=270\backslash$mathrm$\{$~J$\}/(\backslash$mathrm$\{$kg$\}.\backslash$mathrm$\{$K$\})\backslash ))$ leaks out of the spacecraft. How long will it take for the pressure to drop below $50$ kPa $?$ Pressures below $50$ kPa can result in adverse health effects for humans, and thus the time taken provides an estimate of how long the astronaut has, to plug the hole. You can assume that the velocity of the oxygen$-$rich air exiting the hole in $\backslash (300\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ properties $($density$,$ pressure, etc.$)$ of the gas at the exit are the same as that inside the spacecraft, and that the temperature inside the spacecraft remains more or less constant. $[$Hint: the density inside the spacecraft changes with time$]$

$(7$ points$)$

Diameter of hole $\backslash (=4\backslash$mathrm$\{$~mm$\}\backslash )$