Center-of-momentum frame

A center-of-momentum frame (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 necessarily be at rest at the origin of the coordinate system.

Properties

In the centre 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 relativity, COM frame exists for a massive system. 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 (positive invariant mass) of the system.

Systems which have energy but zero invariant mass (such as photons moving in a single direction, or equivalently, plane electromagnetic waves) do not have COM frames, because there is no frame which they have zero net momentum. Because of the invariance of the speed of light, such massless systems must travel at the speed of light in any frame, and therefore always possess a net momentum-magnitude which is equal to their energy divided by the speed of light: p = E/c.

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 %2B m_2v_2}{m_1%2Bm_2}

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^\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 %2B m_2v_2}{m_1%2Bm_2} - \frac{{V_1}{m_1%2B{v_1}m_2}}{m_1%2Bm_2}
V_1^\prime = \frac{m_1v_1 %2B m_2v_2 - m_1v_1 - v_1m_2}{m_1%2Bm_2}
V_2^\prime = V_{CM}
V_2^\prime = \frac{m_1v_1 %2B m_2v_2}{m_1%2Bm_2}

See also