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## Homework Statement

Two identical carts (of mass m) are free to move on a frictionless, straight horizontal track. The masses are connected by a spring of constant k and un-stretched length [itex]l_{0}[/itex]. Initially the masses are a distance [itex]l_{0}[/itex] apart with the mass on the left having a speed [itex]v_{0}[/itex] to the right and the mass on the right at rest. Find the position of mass on the left as a function of time.

## Homework Equations

[itex]q=x_{2}-x_{1}-l_{0}[/itex]

[itex]\dot{q}=\dot{x_{2}}-\dot{x_{1}}[/itex]

[itex]\ddot{q}=\ddot{x_{2}}-\ddot{x_{1}}[/itex]

## The Attempt at a Solution

[itex]m\ddot{x_{1}}=-k(x_{2}-x_{1}-l_{0})[/itex]

[itex]m\ddot{x_{2}}=k(x_{2}-x_{1}-l_{0})[/itex]

remembering [itex]\ddot{q}=\ddot{x_{2}}-\ddot{x_{1}}[/itex]

[itex]\ddot{q}=\frac{2k}{m}q=\omega^{2}q[/itex]

[itex]q(t)=c_{1}e^{\omega t}+c_{2}e^{-\omega t}[/itex]

Just wondering if I'm on the right track? If so I'll do the initial conditions and then solve for [itex]x_{1}(t)[/itex]

Thanks for the help!