Car bumpers are designed to limit the extent of damage to the car in the case of low-veloc...

Car bumpers are designed to limit the extent of damage to the car in the case of low-velocity collisions. Consider a passenger car impacting a concrete barrier while traveling at a speed of 4.0 mph. Model the car as a particle of mass m, and assume that the bumper has a spring element in parallel with a shock absorber so that the overall force exerted by the bumper is , where k, δ, and η denote the spring constant, the spring compression, and the bumper damping coefficient, respectively.

Figure P3.27

Derive the equations of motion for the car during the collision.