Two spherical bodies of masses $2 M$ and $M$ and of radii $3 R$ and $R$ respectively are held at a distance $16 R$ from each other in free space. When they are released, the start approaching each other due to the gravitational force of attraction, then find
(a) the ratio of their accelerations during their motion
(b) their velocities at the time of impact.

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Solution

Taking both the bodies as a system, from conserving momentum of the system.
$m_{1} v_{1} - m_{2} v_{2} = 0 \Rightarrow \frac{m_{1}}{m_{2}} = \frac{v_{2}}{v_{1}} = 2$
again, if the accelerations are $a_{1}$ and $a_{2}$ , the net external force on the system $= 0$
$\Rightarrow {¯ ¯ ¯ a}_{C M} = 0 \Rightarrow m_{1} a_{1} = m_{2} a_{2} = 0$
$\frac{m_{1}}{m_{2}} = \frac{a_{2}}{a_{1}} = 2$
Now conserving the total mechanical energy, we have
$- \frac{G (2 M) M}{16 R} = - \frac{G (2 M) M}{4 R} + \frac{1}{2} (2 M) v_{1}^{2} + \frac{1}{2} (M) v_{2}^{2}$
$v_{1} = \sqrt{\frac{G M}{8 R}}, v_{2} = 2 \sqrt{\frac{G M}{8 R}}$
Note: The velocities and accelerations are w.r.t. the inertial reference frame (i.e. the centre of mass of the system).

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