You are studying the behavior of a car bumper during low velocity crashes. The car is modeled as a mass, and the bumper has a spring and damper (assuming no plastic deformation at low velocity).
When the bumper comes in contact with the barrier, the system is modeled by the differential equation mö + bi + kx = 0 Where x is the position of the car relative to when contact is made (so that c(0) = 0), m is the car mass, b is the bumper damping coefficient, and k is the bumper spring constant.
Suppose m = 1000 kg, b = 2000 N s m-1, and k = 10,000 N m-1, and the initial velocity of the car is c(0) = 2 ms-1.
(a) Find x(t), t > 0.
(b) The solution is only valid for as long as the bumper remains in contact with the wall. For how long is the solution valid?