B Alternative elastic collision formula / physical interpretation

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The discussion centers on the standard formulas for final velocities in elastic collisions and their alternative interpretations. The alternative formulas, derived from rearranging terms, express final velocities in terms of total momentum and average mass, suggesting a connection to the center of mass velocity. It is noted that in the center of mass frame, the velocities of the particles reverse, indicating a time-reversal symmetry in one-dimensional systems. For three-dimensional systems, the momentum conservation principle holds, leading to equal magnitudes of initial and final velocities for both particles. The conversation raises questions about the physical interpretation and complexity of the standard versus alternative formulas.
bobdavis
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Standard formula for final velocities ##v_1##, ##v_2## in elastic collision with masses ##m_1##, ##m_2## and initial velocities ##u_1##, ##u_2## is given by $$v_1 = \frac{m_1-m_2}{m_1+m_2}u_1+\frac{2m_2}{m_1+m_2}u_2$$$$v_2 = \frac{2m_1}{m_1+m_2}u_1+\frac{m_2-m_1}{m_1+m_2}u_2$$.

By rearranging terms this seems to be equivalent to $$v_1 = \frac{p}{\bar{m}}-u_1$$$$v_2 = \frac{p}{\bar{m}}-u_2$$ where ##p = m_1u_1+m_2u_2## is total momentum and ##\bar{m} = \frac{m_1+m_2}{2}## is average mass.

The term ##\frac{p}{\bar{m}}## seems to be the same as ##2v_c## where ##v_c## is the velocity of the center of mass of the system. By substituting this into the formula and rearranging it seems the formula is equivalent to $$\bar{v}_1=v_c$$$$\bar{v}_2=v_c$$ where ##\bar{v}_i=\frac{u_i+v_i}{2}## is the average of the initial and final velocities of particle ##i##

Are these alternative formulas correct and if so is there a good way to physically interpret these alternative formulas and the "momentum over average mass" term? If these formulas are correct is there a reason the seemingly more complex standard formula is used instead?
 
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bobdavis said:
Are these alternative formulas correct and if so is there a good way to physically interpret these alternative formulas and the "momentum over average mass" term?
It is interpreted as Galilean transformed event that in Center of Mass system velocity of each particle does not changes its magnitude but signature by collision.
 
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I see so in the center of mass system ##v_c = 0## and the formula becomes $$\bar{v}_1=\frac{u_1+v_1}{2}=0$$$$\bar{v}_2=\frac{u_2+v_2}{2}=0$$ so $$v_1 = -u_1$$$$v_2=-u_2$$ ?
 
Yes, for 1D system as if a time reverse takes place.
For 3D system in COM system,
m_1\mathbf{u_1}+m_2\mathbf{u_2}=m_1\mathbf{v_1}+m_2\mathbf{v_2}=0
|\mathbf{u_1}|=|\mathbf{v_1}|
|\mathbf{u_2}|=|\mathbf{v_2}|
 
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