Q$.1($So pes$)$

A raitroad car of mass sooo ke travelling at a speed of We$-5.$o m$/$s is stopped at the end of the tracks by a spring and damper mechanism as shown. If the stiffress of the spring is $\backslash (\backslash$mathrm$\{$k$\}=45\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )$ and it is critically damped. Considering t$=0$ when the car touches $($engages with$)$ the spring$-$damper mechanism, and determine:

a$)$ Displacement $\backslash ($ x$($t$)\backslash )$

b$)$ The velocity of the v$($t$)$

c$)$ The time for which the velocity of the car is zero

d$)$ The displacement when the velocity of the car is zero.

e$)$ The displacement and velocity of the car when t$=3$ sec$.$ Innore any loss during the impact of the car on the spring$-$damper mechanism.