X(t) of a diatomic molecule with given v_0

[AbigailM]

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 $l_{0}$. Initially the masses are a distance $l_{0}$ apart with the mass on the left having a speed $v_{0}$ 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

$q=x_{2}-x_{1}-l_{0}$

$\dot{q}=\dot{x_{2}}-\dot{x_{1}}$

$\ddot{q}=\ddot{x_{2}}-\ddot{x_{1}}$

The Attempt at a Solution

$m\ddot{x_{1}}=-k(x_{2}-x_{1}-l_{0})$

$m\ddot{x_{2}}=k(x_{2}-x_{1}-l_{0})$

remembering $\ddot{q}=\ddot{x_{2}}-\ddot{x_{1}}$

$\ddot{q}=\frac{2k}{m}q=\omega^{2}q$

$q(t)=c_{1}e^{\omega t}+c_{2}e^{-\omega t}$

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

Thanks for the help!

 

[Infinitum]

Hi AbigailM!

It would help if you state what x₁ and x₂ are...

 

[AbigailM]

oops, x1 is the position of cart 1 and x2 is the position of cart 2.

 

[Infinitum]

What have you chosen as the origin? I believe q is the elongation/compression of the spring? Is cart 1 the left cart, and cart 2 the right?PS : Its good to state all assumptions before solving the problem