next up previous contents
: Collisions in 1 Dimension : Center of Mass Reference : Center of Mass Reference   目次

Kinetic Energy in the CM Frame


If we extend this treatment to the kinetic energy $E_k$, we also obtain an interesting statement:

\begin{displaymath}
E_{k,\rm tot} = \sum_i \frac{1}{2} m_i v_i^2 = \sum_i
\frac{p_i^2}{2m_i}.
\end{displaymath} (4.38)

Now we recall (from above) that $\vec{v}_i = \vec{v}_i' + \vec{V}_{\rm
cm}$ so that
\begin{displaymath}
\vec{p}_i = m_i\vec{v}_i = m_i\vec{v}_i' + m_i\vec{V}_{\rm cm}
\end{displaymath} (4.39)

and
\begin{displaymath}
E_{ki} = \frac{p_i^2}{2m_i} = \frac{m_iv_i'^2}{2m_i} +
\frac...
...t\vec{V}_{\rm cm}^2}{2m_i} + \frac{m_i^2
V_{\rm cm}^2}{2m_i}.
\end{displaymath} (4.40)

If we sum this:
$\displaystyle E_{k,{\rm tot}}$ $\textstyle =$ $\displaystyle \sum_i E_{ki} = \sum_i \frac{p_i'^2}{2m_i} + \frac{P_{\rm
tot}^2}...
..._{\rm cm}\cdot\underbrace{\sum_i
m_i\vec{v}_i'}_{= \sum_i\vec{p}_i'   =  0}$  
  $\textstyle =$ $\displaystyle E_k({\rm in  cm}) + E_k({\rm of  cm})$ (4.41)

We thus see that the total kinetic energy in the lab frame is the sum of the kinetic energy of all the particles in the CM frame plus the kinetic energy of the CM frame itself (viewed as single ``object''). To put it another way, the kinetic energy of a baseball flying through the air is the kinetic energy of the ``baseball itself'' (the entire system viewed as a particle) plus the kinetic energy of all the particles that make up the baseball measured in the CM frame of the baseball itself. This is comprised of rotational kinetic energy (which we will shortly treat) plus all the general vibrational (atomic) kinetic energy is what we would call heat.

We see that we can indeed break up big systems into smaller systems and vice versa!


next up previous contents
: Collisions in 1 Dimension : Center of Mass Reference : Center of Mass Reference   目次
Robert G. Brown 平成17年1月27日