$2.($a$)$ A jet hits a stationary car body at a speed $\backslash (\backslash$mathrm$\{$V$\}\_\{\backslash$mathrm$\{$J$\}\}\backslash )($Figure b$),$ causing the car body to accelerate. After time t $,$ the car speed becomes $\backslash (\backslash$mathrm$\{$V$\}(\backslash$mathrm$\{$t$\})\backslash ).$ Given the mass of the car body $\backslash (\backslash$mathrm$\{$m$\}\_\{\backslash$mathrm$\{$e$\}\}\backslash )$ the crosssectional area of the jet $\backslash ($ A $\backslash ),$ and the water density $\backslash (\backslash$rho $\backslash ),$ use the Reynolds Transport Theorem $($RTT$)$ form of the momentum equation to express the force F$($t$)$ exerted by the jet on the car body. $($A control volume diagram showing velocity and forces must be drawn.$)$

$($b$)$ Taking the water body in contact with the car body as the moving control volume, and assuming the mass of the control volume is $\backslash (\backslash$mathrm$\{$m$\}\_\{\backslash$boldsymbol$\{\backslash$omega$\}\}\backslash )$ use the RTT form of the momentum equation to express the force $\backslash ($ F$($t$)\backslash )$ on the water body. $($A control volume diagram showing flow velocity and forces must be drawn.$)$

$($c$)$ Using the answers from $($a$)$ and $($b$),$ determine the speed $\backslash (\backslash$mathrm$\{$V$\}(\backslash$mathrm$\{$t$\})\backslash )$ of the car body and the car's acceleration $\backslash (\backslash$mathrm$\{$a$\}(\backslash$mathrm$\{$t$\})\backslash ).$

$($d$)$ Determine the car body's initial acceleration a and terminal velocity $\backslash (\backslash$mathrm$\{$V$\}\_\{\backslash$boldsymbol$\{\backslash$sim$\}\}\backslash )$

$($e$)$ When the car's speed reaches $\backslash ($ V$\_\{\backslash$mathbf$\{$c$\}\}\backslash )$ what is the instantaneous force $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$e$\}\}\backslash )$ exerted on the car body? If you want to maintain a constant speed $\backslash ($ V$\_\{$c$\}\backslash )$ what force $\backslash ($ F$\_\{$s$\}\backslash )$ is required to be applied to the car body?

$($f$)$ In reference to $($e$),$ assuming $\backslash (\backslash$mathrm$\{$m$\}\_\{\backslash$mathbf$\{$w$\}\}\backslash )\backslash (\backslash$mathrm$\{$m$\}\_\{\backslash$mathbf$\{$c$\}\backslash$text $\{$ simplify $\}\}\backslash )$ the force $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$e$\}\}\backslash )$ Compare the "non$-$steady$-$state solution Fe$"$ with the "steady$-$state solution $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$s$\}\}\backslash ).$ What is the difference between the two, and what causes this difference?