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Formulas/physics/Centre Of Mass/Velocity in Centre of Mass Frame

Velocity in Centre of Mass Frame

Velocity of a particle in the CM frame. Total momentum in CM frame is always zero.
Class 11Class JEE
Derivation

The CM frame

The centre of mass frame (or CM frame) is a reference frame that moves with the CM of the system. In this frame, the CM is always at rest.

The velocity of particle ii in the CM frame:

v⃗icm=v⃗i−v⃗cm\vec{v}_i^{cm} = \vec{v}_i - \vec{v}_{cm}

Total momentum in the CM frame is zero

∑miv⃗icm=∑mi(v⃗i−v⃗cm)=∑miv⃗i−v⃗cm∑mi=Mv⃗cm−Mv⃗cm=0\sum m_i \vec{v}_i^{cm} = \sum m_i (\vec{v}_i - \vec{v}_{cm}) = \sum m_i \vec{v}_i - \vec{v}_{cm}\sum m_i = M\vec{v}_{cm} - M\vec{v}_{cm} = 0

The total momentum in the CM frame is always exactly zero. This is the defining property of the CM frame — it is also called the zero momentum frame.

Why the CM frame is useful

In the CM frame:

  • Total momentum is zero — simplifies collision calculations enormously
  • The two bodies in a collision approach each other, collide, and separate
  • Energy considerations are cleaner

Elastic collision in CM frame: Both bodies simply reverse their velocities (since momentum must stay zero and KE must be conserved).

Two-body collision in CM frame

For two bodies with masses m1m_1, m2m_2 and velocities u1u_1, u2u_2:

vcm=m1u1+m2u2m1+m2v_{cm} = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

Velocities in CM frame:

u1cm=u1−vcm=m2(u1−u2)m1+m2u_1^{cm} = u_1 - v_{cm} = \frac{m_2(u_1-u_2)}{m_1+m_2}

u2cm=u2−vcm=−m1(u1−u2)m1+m2=−m1m2u1cmu_2^{cm} = u_2 - v_{cm} = \frac{-m_1(u_1-u_2)}{m_1+m_2} = -\frac{m_1}{m_2}u_1^{cm}

In the CM frame, the momenta are equal and opposite: m1u1cm=−m2u2cmm_1 u_1^{cm} = -m_2 u_2^{cm}. ✓

Converting back to lab frame

After finding velocities in the CM frame, add v⃗cm\vec{v}_{cm} to convert back:

v⃗ilab=v⃗icm+v⃗cm\vec{v}_i^{lab} = \vec{v}_i^{cm} + \vec{v}_{cm}

Remember
The CM frame is most useful for elastic collisions and for finding the minimum kinetic energy of a system. The KE in the CM frame is the "internal" KE — it is what can be converted to other forms in a collision, while the CM's translational KE is always conserved (since CM velocity doesn't change in the absence of external forces).