$2\mathrm{.}43.$ A certain jet$-$powered ground vehicle is subjected to a nonlinear drag force. Its equation of motion, in British Engineering units, is

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$50\backslash$hat$\{$v$\}=$f$-\backslash$left$(20$ v$+0\mathrm{.}05$ v$^\{2\}\backslash$right$)$

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$($a$)$ Find the vehicle's steady$-$state speed if the jet's force $\backslash ($ f $\backslash )$ is a step function of magnitude $8000$ lb $.$

$($b$)$ Linearize the model about the reference speed $\backslash ($ v$=0\backslash ),$ and use the time constant of the linearized model to estimate how long it will take the vehicle to reach steady$-$state speed.

$($c$)$ Repeat part $($b$)$ using the steady$-$state speed as the reference speed. Compare the prediction with that of part $($b$).$

$($d$)$ Can you suggest another reference speed that might give a more accurate prediction of the time required to reach steady state?