- Center of momentum frame
A

**center of momentum frame**(or zero-momentum frame, or COM frame) of a system is anyinertial frame in which thecenter of mass "is at rest" (has zero velocity). Note that the "center of momentum" of a system is not a location, but rather defines a particular inertial frame (a velocity and a direction). Thus "center of momentum" already means "center of momentum**frame**" and is a short form of this phrase.A special case of the center of momentum frame is the

**center of mass frame**: an inertial frame in which the center of mass (which is a physical point) is at the origin at all times. In all COM frames, the center of mass is at rest, but it may not be at rest at the origin of the coordinate system.**Properties**In a center of momentum frame the total linear momentum of the system is zero. Also, the total energy of the system is the "minimal energy" as seen from all possible

inertial reference frame s. In the COM frame, the total energy of the system is the "rest energy", and this quantity (when divided by the factor c^{2}) therefore gives therest mass orinvariant mass of the system.**Example problem**An example of the usage of this frame is given below - in a two-body elastic collision problem.The transformations applied are to take the velocity of the frame from the velocity of each particle:

$V\_1^\{prime\}\; =\; V\_1\; -\; V\_\{CM\}$

where $V\_\{CM\},$ is given by:

$V\_\{CM\}\; =\; frac\{m\_1v\_1\; +\; m\_2v\_2\}\{m\_1+m\_2\}$

If we take two particles, one of mass m

_{1}moving at velocity V_{1}and a second of mass m_{2}, then we can apply the following formulae::$V\_1^\{prime\}\; =\; V\_1\; -\; V\_\{CM\}$

:$V\_2^\{prime\}\; =\; -\; V\_\{CM\}$

After their collision, they will have speeds:

:$V\_1^\{prime\}\; =\; V\_\{CM\}\; -\; V\_1$

:$V\_1^\{prime\}\; =\; frac\{m\_1v\_1\; +\; m\_2v\_2\}\{m\_1+m\_2\}\; -\; frac$v_1}{m_1+m_2{m_1+m_2}

:$V\_1^\{prime\}\; =\; frac\{m\_1v\_1\; +\; m\_2v\_2\; -\; m\_1v\_1\; -\; v\_1m\_2\}\{m\_1+m\_2\}$

:$V\_2^\{prime\}\; =\; V\_\{CM\}$

:$V\_2^\{prime\}\; =\; frac\{m\_1v\_1\; +\; m\_2v\_2\}\{m\_1+m\_2\}$

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