Two particles having masses m₁ and m₂ start moving towards each other from the state of rest from infinite separation. Their relative velocity of approach when they are interacting gravitationally at a separation r will be

$\sqrt{\frac{2G\left(m_1+m_2\right)}{r}}$

CONCEPT OF REDUCED MASS (µ)

Let m, be at rest and think that m₂ has been replaced by µ and is moving with velocity v. Then by Law of Conservation of Energy

$\frac{1}{2} m_1(0)^2+\frac{1}{2} \mu v^2=-\frac{Gm_1 m_2}{r}+0$

where, µ = reduced mass of system = $\frac{m_1 m_2}{m_1+m_2}$

$\Rightarrow v=\sqrt{\frac{2 G\left(m_1+m_2\right)}{r}}$