# Classical Mechanics: Two Body Problem

## Homework Statement

Consider two objects with masses ##m_1## and ##m_2## exerting forces on each other with magnitude ##F##. If no other net forces act on the objects, they obey the equations of motion

##m_1\ddot r_1=F##, ##m_2\ddot r_2=-F##

Show that the corresponding equations of motion for the centre of mass ##R## and the relative position ##r## of the two objects in terms of the total mass ##M## and the reduced mass ##\mu##, are given by

##M\ddot R=0##, ##\mu\ddot r=F##

and that

##r_1=R+\frac{m_2}{M}r##, ##r_2=R-\frac{m_1}{M}r##

## The Attempt at a Solution

Using the relation where ##F_{12}+F_{21}=0## I said ##m_1\ddot r_1+m_2\ddot r_2=0##. This equates to ##M\ddot R=0## as asked.

To show ##\mu\ddot R=F##, I manipulated ##m_1a_1=-m_2a_2## to get ##a_2=-\frac{m_1}{m_2}a_1##. Then using

##a_{rel}=a_1-a_2=a_1+\frac{m_1}{m_2}a_1=(1+\frac{m_1}{m_2})a_1##

##=\frac{m_2+m_1}{m_1m_2}m_1a_1##, Therefore ##\frac{F}{\mu}=a_{rel}=\ddot r##

And finally ##\mu\ddot r=F##

I can't find a way of finding ##r_1## and ##r_2## as asked however. I tried using ##m_1r_1+m_2r_2=0## but I can't find it as I'm asked. Any help here would be greatly appreciated.