# Center of momentum frame

Center of momentum frame

A center of momentum frame (or zero-momentum frame, or COM frame) of a system is any inertial frame in which the center 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 frames. In the COM frame, the total energy of the system is the "rest energy", and this quantity (when divided by the factor c2) therefore gives the rest mass or invariant 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:

$V_1^\left\{prime\right\} = V_1 - V_\left\{CM\right\}$

where $V_\left\{CM\right\},$ is given by:

$V_\left\{CM\right\} = frac\left\{m_1v_1 + m_2v_2\right\}\left\{m_1+m_2\right\}$

If we take two particles, one of mass m1 moving at velocity V1 and a second of mass m2, then we can apply the following formulae:

:$V_1^\left\{prime\right\} = V_1 - V_\left\{CM\right\}$

:$V_2^\left\{prime\right\} = - V_\left\{CM\right\}$

After their collision, they will have speeds:

:$V_1^\left\{prime\right\} = V_\left\{CM\right\} - V_1$

:$V_1^\left\{prime\right\} = frac\left\{m_1v_1 + m_2v_2\right\}\left\{m_1+m_2\right\} - frac$v_1}{m_1+m_2{m_1+m_2}

:$V_1^\left\{prime\right\} = frac\left\{m_1v_1 + m_2v_2 - m_1v_1 - v_1m_2\right\}\left\{m_1+m_2\right\}$

:$V_2^\left\{prime\right\} = V_\left\{CM\right\}$

:$V_2^\left\{prime\right\} = frac\left\{m_1v_1 + m_2v_2\right\}\left\{m_1+m_2\right\}$

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