Consider two satellites A and B of equal mass m, moving in the same circular orbit of radius r around the earth E but in opposite sense of rotation and therefore on a collision course (see figure)
(i). In terms of `G,M_(e)` m and r find the total mechanical energy `E_(A)+E_(B)` of the two satellite plus earth system before collision
(ii). If the collision is completely inelastic so that wreckage remains as one piece of tangled material (mass=2m) find the total mechanical energy immediately after collision.

Answer

(i). `(-GmM_(e))/(r)`
(ii). `-(2GmM_(e))/(r)`
(iii). Free fall

Explanation

By applying conservation of linear momentum
`m(v)-m(v)=2mv',v'=0`
Initially energy of a satellite 'A' and 'B' is
`E_(A)=-(GM_(e)m)/(2R),E_(B)=-(GM_(e)m)/(2R)`
Total enegy `E_(A)+E_(B)=-(GM_(e)m)/(R)`
After collision velocity of satellite becomes zero then the K.E.=0 therefore total mechanical energy becomes
`-(GM_(e)2M)/(R)`