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In dynamics when doing collisions from the centre of masses' frame of reference we use reduced mass. What is is it and why do we use it?

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Any time you have an equal and opposite force acting on two different masses, the notion of reduced mass arises.

Consider some interaction with force $F$ acting on two masses $m_1$ and $m_2$. The motion of each mass is affected by

$$ \begin{aligned} a_1 & = -\frac{F}{m_1} & a_2 & = +\frac{F}{m_2} \end{aligned}$$

The relative motion is

$$ a_2 - a_1 = \frac{F}{m_2} + \frac{F}{m_1} = F \left( \tfrac{1}{m_1} + \tfrac{1}{m_2} \right) $$

Solve the above in terms of $F$ to get

$$ F = \underbrace{ \left( \tfrac{1}{m_1} + \tfrac{1}{m_2} \right)^{-1} }_{\text{reduced mass}} \left(a_2-a_1\right) $$

Note that $\left( \tfrac{1}{m_1} + \tfrac{1}{m_2} \right)^{-1} = \frac{m_1 m_2}{m_1 + m_2} $.

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