Ratio of Kinetic Energies in Center of Mass Frame

[tanzl]

Homework Statement

Consider a system of two particles, with masses m1 and m2. What is
the ratio of their kinetic energies, T1=T2, in the center-of-mass frame?

The Attempt at a Solution

I do not really understand the question because center of mass frame was not taught in my class. From wikipedia, "center of mass frame is defined as being the particular inertial frame in which the center of mass of a system of interest is at rest (has zero velocity)". What is that supposed to mean? For example, in a collision of two particles (A and B), the center of mass is refer to the center of mass of A or B?

From wikipedia also, "In this special inertial frame where the center of mass is at rest, the total linear momentum of the system is zero." Can anybody explain this? Thanks...

 

[Chi Meson]

From wikipedia, "center of mass frame is defined as being the particular inertial frame in which the center of mass of a system of interest is at rest (has zero velocity)".

A system is any group of objects that you decide to examine. They don't have to be attached, but you have to always keep all the objects in mind (and in the math) as you solve your problem.

Let's say that we have the system of two objects and "2" is twice the mass of "1." No matter where either one is, if you connected them with a massless rod, there would be a "balance point" on that rod that would be closer to the more massive object. That's the center of mass of the system.

If both objects are moving, then the center of mass point would move also. In this case, it would move so it would always be 1/3 (of total separation distance) near to the more massive object, and 2/3 to the less massive ("1" is twice as far from CM as "2").

If the observer stayed at this point and called it his reference point, then no matter what motion the two objects did, from the CM reference point, the objects would only be moving apart, or moving together, with exactly the same but opposite momentum. The less massive object, "1," would have to move with twice relative velocity in order to keep it 2x as far from the CM as the twice as massive object.

That means, from the CM, if "1" and "2" have the same magnitude of momentum, but "2" has twice the mass, what does that say about their velocities, and therefore their KEs?

Edit: We are assuming a non-rotating system here.

 

[tanzl]

By Galilean transformation, I am able to obtain velocity of A and B in the center of mass frame.
Then, by plugging in the two expression into kinetic energy formula, the ratio I got is
$$\frac{T_1}{T_2}$$=$$\frac{m_2(v_1-v_2)^2}{m_1(v_2-v_1)^2}$$
Since (v₁-v₂)²=(v₂-v₁)²

The ratio equals m₂/m₁