Spring Oscillations Problem Question

[GoGoGadget]

Homework Statement

You have a job at a medical forensics lab investigating an accident at a commuter railroad station. Your task is to determine the response of the safety system that prevented a railroad car from crashing into the station. Because the brakes on the passenger car failed, it could not stop. The safety system at the end of the track was a large horizontal spring with a hook that grabbed onto the car when it hit preventing the car from crashing into the station platform. To determine the cause of passenger injuries, you want to know the frequency and amplitude of the car's oscillation after it hit the spring based on the specifications of the passenger car, the specifications of the spring, and the speed of the passenger car.

Homework Equations

See attached document.

The Attempt at a Solution

See attached document for full problem with solution. It was an example my professor gave. I'm unsure though of how you would find the amplitiude (A) and frequency (f) in the final equation though. I understand the differential equations to it but I didn't understand my professor's problem solving method for the final part of the problem. In the attached document, he relates the general equation of oscillation x = a Sin (bt + c) to aspects of the picture and finds equations for the unknown to finally find everything algebraically. My question, for how he finds b, he takes V/A of the problem. I really didn't understand that. I know b typically relates to frequency but I'm not understanding what he did with the V/A part of the problem. Otherwise, for f, it looked like he took the equation for finding the period (T), with equation T = 1/2TT to incorporate it into finding the frequency for the problem. If anyone would be able to take a look at this to provide input, it'd be greatly appreciated.

 

[haruspex]

You're ok with the equation for v(t), right? And that the max value of that is A√(k/m)? And that this must equal V?