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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:

$$E=(\frac{1}{2}m_1v_1^2)+(\frac{1}{2}m_2v_2^2)=(\frac{1}{2}m_1v_1'^2)+(\frac{1}{2}m_2v_2'^2)\tag{1}$$

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)

$$ε_1'=\frac{1}{2}m_1V_1'$$and$$ε_2'=\frac{1}{2}m_2V'_2\tag{2}$$

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?

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  • $\begingroup$ Welcome New contributor user7852656! I've flagged your question for moderator attention due to its similarity to your closed (as a duplicate) and deleted question here $\endgroup$ – Alfred Centauri Oct 18 '18 at 2:37
  • $\begingroup$ Thanks. I think I am very close to really getting it. But some pieces do not really make sense quite yet. $\endgroup$ – user7852656 Oct 18 '18 at 2:42
  • $\begingroup$ If this question is really different from the previous one, you should not use the same title. Let each question be different and separate. If you have many doubts about one topic, write separate and different questions for each one of them - again do not repeat the title. Since I cannot comment, can some mod or someone with power change this answer into a comment. Thx. $\endgroup$ – Ricardo Magallanes Oct 18 '18 at 3:11
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    $\begingroup$ Possible duplicate of Why is the laboratory frame energy always greater than the center of mass frame energy? $\endgroup$ – Aaron Stevens Oct 18 '18 at 3:45
  • $\begingroup$ I'm afraid I do not see how this question is different from your other question. You say "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)." - but that is precisely what your previous question asked about, too: ""Why is the laboratory frame energy always greater than the center of mass frame energy during collisions? $\endgroup$ – ACuriousMind Oct 18 '18 at 16:31
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The simplest way to acquire an intuition about different frames is to visualize a traveling train. Everything in the train is moving with the train velocity, but as long as there is no acceleration or deceleration, an experiment done in a train is done in the center of mass frame of the train, i.e. the sum of momenta within the train is zero. Suppose you scatter two identical balls with equal and opposite momenta in the train. The total momentum of the two balls will be zero, and thus they will also be in their center of mass frame. BUT

for an observer outside the train, the effect of the train's velocity has to be added to the velocity of the individual particles in the train: to describe the experiment in the laboratory frame ( observer watching moving train) one would have to add the velocity of the train to all three vectors in the experiment, making the description more complicated, and destroying the "sum of momenta of the balls equals zero".

That the center of mass frame has less energy than the laboratory is self evident, since less momentum is take into account, as common to all objects in the problem.

The same holds for four vectors, particles decaying in a train for example have simpler expressions and less energy than that given by calculating from the lab ( observer measuring experiment in train), but four vector algebra has to be used.

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