# Conceptual question about rotational and translational kinetic energy

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 kinetic energy and $$W$$ is the work due to friction (F) between the tires and the road surface (assume that's the only external force at play), therefore:

$$\frac 1 2 mv^2 + \frac 1 2 I\omega^2 = \int F\,ds + \int \tau\,d\theta$$

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?

$$\frac 1 2 mv^2 = \int F\,ds$$

$$\frac 1 2 I\omega^2 =\int \tau\,d\theta$$

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..

• I think it is fine because the total energy is the addition of its two components, which are integrated over their own domain, that is independent of each other. – r13 May 1 at 15:50

then the contributory effect of this force is two parts one part imparting negative acceleration and the other part which is the negative torque resisting car spin, $$T=F_a*da \$$ with "da" being the distance of the tire to the CG of the car.
$$F_adt=-(m \alpha+I*da)dt$$