$4\mathrm{.}8.$ Figure P$4\mathrm{.}8$ a shows a car$-$barrier crash model for a car with rear$-$wheel drive. Such a model was used to investigate and improve car designs to protect passengers in a head$-$on collision $[$Kamal$,1982].$ The three significant masses in the model are $m_1,$ the car body exclusive of the engine and the front suspension; $m_2,$ the front suspension; and $m_3,$ the engine and transmission. The elastic elements modeled in the study were $k_1,$ due to the torque box $($that part of the engine crossmember back to the side rails$)$; $k_2,$ due to the front frame $($including the bumper and its brackets$)$; $k_3,$ due to the drive train; and $k_4,$ due to the sheet metal of the fender, hood, and grille. Also, $k_5,$ due to the firewall; $k_6,$ due to the radiator; $k_7,$ due to the engine mounts; and $k_8,$ due to the transmission mounts.

$($a$)$ Derive the equations of motion.

$($b$)$ There is no input in this system, so the initial conditions must be chosen to model the impact with the wall. How many initial conditions must be specified, and what should their values be if the car impacts the wall at $30$ miles per hour?