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My real life problem is to calculate initial translational and angular velocities of a vehicle in a loss of control to a stop (the vehicle will translate and rotate about it's center of mass.)

Initial strategy is to use energy-work theorem, therefore:

$K = W$

where $K$ is the initial kinectic energy and $W$ is the work due to friction (F) between the tires and the road surface (assume thats the only external force at play), therefore:

$mv^2/2 + I\omega^2/2 = \int F\,ds + \int \tau\,d\theta$

Assume I know how to calculete the RHS of the equation. The problem with this, is that i have two variables($v$ and $\omega$) with only one equation.

The question is: Is it valid to write two different equations, one for rotation and the othe for translation, as following?

$mv^2/2 = \int F\,ds$

$I\omega^2/2 =\int \tau\,d\theta$

This way i would be able to solve for $v$ and $\omega$, but I am not so sure i can do it without violating some underliying principle..

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The question is: Is it valid to write two different equations, one for rotation and the othe for translation, as following?

Yes. The change in rotational KE and translational KE are independent of one another. So you can separate them.

But regarding your first equation, note that the work energy theorem states that the net work done on an object equals its change in KE. So you need to specify the initial and final conditions. In your case, the assumption is the vehicle starts out at rest and that $v$ and $\omega$ are the final translational and angular velocities.

Hope this helps.

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