Dynamics collisions - reduced mass

Question

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?

Answer 1

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} $.