A railroad car of mass \(m\) is attached to a stop in a railroad yard. The stop consists of four identical metal rods of length \(L\), radius \(R\) and elastic modulus \(E\) that are arranged symmetrically and are fixed to a rigid wall at one end and welded to a rigid plate at the other. The plate is hooked to the stationary railroad car as shown. In a docking maneuver, a second car of mass \(m\) approaches the first at speed \(v_{1}\). If the second car locks onto the first upon contact, determine the response of the two car system after docking.

Answer: To find the final velocity (vf) of the combined cars, follow these steps: 1. Calculate the initial kinetic energy (K.E.) of the second car: \(K.E. = \frac{1}{2}mv_{1}^{2}\) 2. Determine the spring constant k of a single rod: \(k = \frac{EA}{L}\) and \(A = \pi R^2\) 3. Calculate the equivalent spring constant of the four-rod system: \(k_{eq} = 4k\) 4. Find the maximum compression x in the rods by equating the potential energy in the rods to the initial kinetic energy of the second car: \(P.E. = \frac{1}{2}k_{eq}x^2 = K.E.\) 5. Determine the final velocity vf using conservation of momentum: \(mv_1 = 2mv_f\), solve for vf. By following these steps and substituting the given values, you can calculate the final velocity (vf) of the two-car system after the energy absorption by the rods.