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*From an inertial frame in space, we watch two identical uniform spheres fall toward one another owing to their mutual gravitational attraction. Approximate their initial speed as zero and take the initial gravitational potential energy of the two-sphere system as [itex]U_i[/itex]. When the separation between the two spheres is half the initial separation, what is the kinetic energy of each sphere?*

My solution was this:

Since the mechanical energy of the two-sphere system will be conserved,

[tex]K_i + U_i = K_f + U_f[/tex]

Let K be the kinetic energy of one of the spheres in the final configuration of the system. Since both spheres are identical, K is also the kinetic energy of the other sphere, making the net kinetic energy 2K. Bearing in mind that the kinetic energy of each sphere was given as initially zero, this means:

[tex]U_i = 2K + U_f[/tex]

[tex]\frac {U_i - U_f}{2} = K[/tex]

The gravitational potential energy of such a system is given by:

[tex]U = -\frac{GMm}{r}[/tex]

Let R be the distance between the two spheres at the system's initial configruation, then R/2 is the distance between the two spheres in the system's final configuration, and:

[tex]\frac{-\frac{GMm}{R} - -\frac{GMm}{R/2}}{2} = K[/tex]

[tex]\frac{-\frac{GMm}{R} + \frac{2GMm}{R}}{2} = K[/tex]

[tex]\frac{\frac{GMm}{R}}{2} = K[/tex]

[tex]\frac{-U_i}{2} = K[/tex]

The book's answer is [itex]U_i / 4[/itex], not [itex]-U_i / 2[/itex]. Where is my mistake?