Consider a system of two toy rail cars (i.e., frictionless masses). Suppose that car 1 has mass [math] and is traveling at [math] toward the other car. Suppose car 2 has mass [math] and is moving toward the other car at [math]. There is a bumper on the second rail car that engages at the moment the cars hit and does not let go (it connects the two cars). The bumper acts as a spring with spring constant [math]. The second car is [math] from a wall.

Let [math] be the time that the cars link up. Let [math] be the displacement of the first car from its position at [math], and let [math] be the displacement of the second car from its original position.
 
  1. Set up a system of second-order differential equations that models this situation.
    [math]
    [math] [math]
    [math]
    [math] meters
    [math] meters
    [math] meters/second
    [math] meters/second

  2. Find the solution to this system of differential equations.
    [math] meters
    [math] meters

  3. Will the cars be moving toward the wall, away from the wall, or will they be nearly stationary?