Problem $\backslash (3[30\backslash )$ pts$]$: A $4,000$ kg truck is driving down the hill with an initial velocity of $\backslash ($ v$\_\{1\}\backslash )$ when it sees a $\backslash (1,200\backslash$mathrm$\{$~kg$\}\backslash )$ car stopped in front of it$.$ The truck slams on the brakes while there are $20$ m between the vehicles. After impact, the car slides $30$ m $.$ The coefficient of resittution between the vehicles is $0\mathrm{.}65.$ The static and kinetic coefficients of friction between all tires and the road are $\backslash (\backslash$mu$\_\{$s$\}=0\mathrm{.}4\backslash )$ and $\backslash (\backslash$mu$\_\{$k$\}=0\mathrm{.}25\backslash ).$ Assume that all wheels fully lock up and do not rotate.

$($a$)$ What is the velocity of the car immediately after impact?

$($b$)$ What was the initial velocity $\backslash ($ v$\_\{1\}\backslash )$ of the truck before it applies the brakes?

Please write final answers below

In the workspace below, demonstrate the problem$-$solving process including approach, governing equations and diagrams as needed. You must include your governing equations and diagrams for full credit.