Internal kinetic energy and center of mass kinetic energy

Question

For a given system, how can you tell which one is kinetic energy for center of mass and which one is internal kientic energy?

K = Kcm + K int

For example, "A 150 g trick baseball is thrown at 63 km/h. It explodes in flight into two pieces, with a 39 g piece continuing straight ahead at 85 km/h. How much energy do the pieces gain in the explosion?"

I found the answer using conservation of momentum. I solved for the velocty and then taking the difference in kinetic energies before and after, the change was .984 J. However, relative to the equation, how do I know which one is internal kinetic energy and which is center of mass kinetic energy. Does K in the equation mean to total energy? How would you find Kcm and Kint? Can someone please clarify?

Answer 1

The center of mass KE is equal to the KE of the total mass of the system moving at the center of mass speed. The total KE in a stationary frame of reference is the sum of this center of mass KE and the KE of different masses relative to the center of mass frame of reference. In mathematical terms,

\begin{equation} KE_{lab}^{total} = KE_{lab}^{CM} + \sum\limits_{i=1}^n KE_{CM}^i \end{equation}

Here the subscript refers to the frame of reference while the superscript is the KE of that species. Hence the term $K_{int}$ refers to the KE of individual masses of the system relative to the center of mass, while $K_{CM}$ is the kinetic energy of center of mass as observed from a stationary frame. The kinetic energy lost during the explosion is the difference between the initial and final $KE_{lab}^{total}$. The center of mass kinetic energy can be calculated by knowing the total mass and the velocity of center of mass in the stationary frame. To calculate the internal kinetic energy, one needs to calculate the velocities of individual masses relative to the center of mass i.e $v_{lab}^{i} - v_{CM}$ (vector subtraction).