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39. (a) The momentum of the two-star system is conserved, and since the stars have the
same mass, their speeds and kinetic energies are the same. We use the principle of
conservation of energy. The initial potential energy is
U
i
=
−
GM
2
/r
i
, where
M
is the mass
of either star and
r
i
is their initial center-to-center separation. The initial kinetic energy is
zero since the stars are at rest. The final potential energy is
U
f
=
−
2
GM
2
/r
i
since the final
separation is
r
i
/2. We write
Mv
2
for the final kinetic energy of the system. This is the sum
of two terms, each of which is ½
Mv
2
. Conservation of energy yields

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