Figure 6.44 shows a model of an automobile including the masses of the automobile, seat and body, and head and neck of the passenger. The suspension system of the automobile is modeled by two linear springs \(k_{1}\) and \(k_{2}\) with the wheels assumed to have negligible mass. The head and neck are connected to the seat by a linear spring \(\left(k_{3}\right)\) and a torsional spring \(\left(k_{t}\right)\), while the seat and body is connected to the automobile using a linear spring \(k_{4}\). Derive the equations of motion of the system when the automobile hits a rigid wall at a linear velocity \(v\). Assume that the head is at a height of \(h\) above the mass \(m_{2}\) concentrated at the pivot point \(P\) when the automobile is at rest. Discuss the suitability of the model for predicting the injury to the head during the impact.

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Rigid Wall Head and neck, mi -k3 k4 Seat and body, m Automobile, my k k www. FIGURE 6.44 Modeling of an automobile hitting a rigid wall.