I looked through lots of sources to answer the question, 'Why is lab frame energy (total energy) always greater than the center of mass frame energy?' Many of them provided lots of mathematical answers. Could someone explain this in terms of physics or simple concepts or math invovled?
Difference from the previous question below
In addition to the previous questions, some aspects remain a bit confusing to me. (I decided to leave the previous question as it is, since the existing answers answered the original questions) To highlight my confusion, I am very confused by why the total energy for the lab frame is greater if the lab frame and center of mass frame describe the same event but differ by the frame of reference (the center of the two particles vs. one of the particles). Especially they are under the same assumptions (e.g conservation of momentum and elastic collisions). Is this because the vectors cancel for the center of frame energy when the masses collide?
Due to the conservation of energy, the total energy in the center of mass frame, $E$ is the following:
where $m_1$ is one of the masses, $m_2$ is the other mass, $v_1$ and $v_2$ are initial velocities and $v'_1$ and $v'_2$ are final velocities.
Similarly, the total energy in the lab frame is $ε$ is
$$ε=ε_1'+ ε_2'$$(where $ε_1'$ is the energy of one of the masses in the final state and $ε_2'$ is the other mass in the final state)
where $V_1'$ and $V_2'$ are the final velocities.
From here, how do we end up with $ε=a \times E$? (showing that $ε$ is greater than $E$?)
What do we know about the relationship between Eq1 and Eq2 to determine that ε is greater than E?