Formula for acceleration between to object.

[ManishR]

I don't follow that. I was assuming that your scenario was for three objects, the earth, and two objects in Earth free fall. If that is the case then you need to include all three objects in the same frame. If that is not the case then can you be more specific about what your scenario is?

no, the scenario has been changed, sorry for not mentioning.

what i meant by that is,
let say there is another inertial frame f (beside the CoM, which is now an inertial frame).
if both objects are accelerating in same direction with respect to f,
=> there is an acceleration between CoM and f.
=> CoM cannot be inertial frame (now). because we have assumed that f is an inertial frame. so one of them has to non inertial, CoM is non inertial.

there is only two object in the space.

 

[K^2]

i have no idea what you just said.

however

if there is only two point mass objects m_a & m_b with \vec{r_{0}} where \hat{r_{0}} is from m_a to m_b.

\frac{d^{2}}{dt^{2}}\vec{r_{a}}=-\left(\frac{m_{b}}{m_{a}+m_{b}}\right)\frac{d^{2}}{dt^{2}}\vec{r_{0}}

where \vec{r_{a}} is vector position of m_a from an inertial frame.

Yes... And that tells you that you cannot solve the problem from accelerated frame how? Your dr_a/dt will be completely different, but dr/dt will be exactly the same, since it is frame-invariant.

 

[ManishR]

And that tells you that you cannot solve the problem from accelerated frame how?

i have never said you cannot solve problem from accelerated frame.
what i said is,

a_a = \frac{d^2r_a}{dt^2} = -\frac{F_a}{m_a} = -G\frac{m_b}{(r_a + r_b)^2}

the step need correction if you use non inertial frame.

 

[K^2]

Ah, of course. Acceleration of a body depends on acceleration of the reference frame. But we know how to correct for that, so we can still work from accelerated reference frame.