Problem \#1 - Rail car buffer In this problem we will use some of the ideas from last week's homework and this week's lectures in an engineering problem. Suppose that we are to design a railroad car buffer. The purpose of the buffer is to stop the motion of a free-rolling railroad car, in a short distance, but with a low maximum force on the car. The buffer, shown below, consists of a spring and damper. The spring stiffness is $K=90,000$ Newton/meter, and the damping coefficient is $B=60,000$ Newton-sec/meter. The mass of the railroad car, when loaded is $M=10,000$ kg. Assume that the car collides with the buffer with an initial speed $V_{0}=2$ meter/second, and that the car remains attached to the buffer once the collision has occurred at $t=0$. Part 1 - One Car a) Determine the differential equation for the displacement of the car after the collision. b) Is this system overdamped, underdamped, or critically damped? c) Plot the roots of the characteristic equation in the complex plane (imaginary vs real axes). Indicate the root locations with an "x." d) What are the initial conditions of the system? e) Solve the differential equation for the displacement, and plot the solution. f) When will the car return to the point of impact? g) Find the expression for the total force on the car, $f_{m}(t)=\ldots$ Plot this force. h) What is the maximum (absolute value) force on the car $[\mathrm{N}]$ ?

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