Consider a system of two toy rail cars (i.e., frictionless masses). Suppose that car 1 has mass $2 \ \mathrm{kg}$ and is traveling at $3 \ \mathrm{m/s}$ toward the other car. Suppose car 2 has mass $1 \ \mathrm{kg}$ and is moving toward the other car at $9 \ \mathrm{m/s}$. 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 $2 \ \mathrm{N/m}$. The second car is $10 \ \mathrm{m}$ from a wall. Let $t = 0$ be the time that the cars link up. Let $x_1$ be the displacement of the first car from its position at $t = 0$, and let $x_2$ 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.
$\boldsymbol{\vec{x}^{\, \prime\prime}} =$
 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$
$\boldsymbol{\vec{x}}$
$x_1(0) =$ meters
$x_2(0) =$ meters
$x_1^{\,\prime}(0) =$ meters/second
$x_2^{\,\prime}(0) =$ meters/second

2. Find the solution to this system of differential equations.
$x_1(t) =$ meters
$x_2(t) =$ meters

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