Two identical planets each of mass m are approaching towards each other due to their mutual gravitational field. If the speed of one planet relative to the other is v, the K.E. of the system is equal to

(1/4) mv²

The K.E. of the system of two mass system moving under their mutual interaction can be given as KE = $(1 / 2) \mu v_{\text {res }}^2$

where $\mu=\frac{m_1 m_2}{m_1+m_2}$  &  $v_{\text {velocity }}=\left(v_1+v_2\right)$

$\Rightarrow \mu=\frac{m m}{m+m}=\left(\frac{m}{2}\right) v^2=\frac{m v^2}{4}$