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Archimedes' Principle

Two particles...

Question

Two particles of mass $1 k g$ and $0.5 k g$ are moving in the same direction with speed of $2 m / s e c$ and $6 m / s e c$ respectively on a smooth horizontal surface. Find the speed of centre of mass of the system.

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Solution

Velocity of c.m of the system is given as
${\to v}_{c m} = \frac{m_{1} {\to v}_{1} + m_{2} {\to v}_{2}}{m_{1} + m_{2}}$
Since the particles $m_{1}$ and $m_{2}$ are moving unidireactionally
$m_{1} {\to v}_{1}$ and $m_{2} {\to v}_{2}$ are parallel
$\Rightarrow ∣ ∣ m_{1} {\to v}_{1} + m_{2} {\to v}_{2} ∣ ∣ = m_{1} v_{1} + m_{2} v_{2}$
Therefore $v_{c m} = \frac{m_{1} {\to v}_{1} + m_{2} {\to v}_{2}}{m_{1} + m_{2}} = \frac{| m_{1} v_{1} + m_{2} v_{2} |}{m_{1} + m_{2}} = \frac{(1) (2) + (1 / 2) (6)}{(1 + \frac{1}{2})} m / s = 3.33 m / s e c$

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