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COM in Collision and Explosion

Two particles...

Question

Two particles of mass $M_{A}$ and $M_{B}$ and there velocities are $V_{A}$ and $V_{B}$ respectively collides. After collision they inter changes their velocities then ratio of $\frac{M_{A}}{M_{B}}$ is?

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Solution

Before collision momentum ${= \to ρ}_{1} = M_{A} V_{A} + M_{B} V_{B}$
after collision momentum u ${\to ρ}_{2} = M_{A} V_{B} + M_{B} V_{A}$
momentum is conserve ${\to ρ}_{1} = {\to ρ}_{2}$
$M_{A} V_{A} + M_{B} V_{B} = M_{B} V_{A} + M_{A} V_{B}$
divide both side by $M_{B}$
$\frac{M_{A}}{M_{B}} V_{A} + V_{B} = V_{A} + \frac{M_{A}}{M_{B}} V B \frac{M_{A}}{M_{B}} (V_{A} - V_{B}) = (V_{A} - V_{B})$
clear that $\frac{M_{A}}{M_{B}} = 1$

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