Masses colliding due to gravitational attraction

[jolly_math]

Homework Statement

Consider two identical masses that interact only by gravitational attraction to each other. If one mass is fixed in place and the other is released from rest, then the two masses collide in time T. If both masses are released from rest, they collide in time
(A) T/4
(B) T/(2√2)
(C) T/2
(D) T/√2
(E) T

Relevant Equations

F=Gm1m2/(r^2)
KE = 1/2(mv^2)

When one mass is held fixed, the other mass acquires a speed v from gravity.
I don't understand the following explanation:
When both masses can move, they share the kinetic energy, so both have speed v/√2, so the relative speed is √2v. Hence to collapse the same distance r, the latter case will be √2 times faster, thus the time will be T/√2.

When one mass is held fixed, there an external force applied to prevent it from moving, but it is not applied over a distance - is this why energy is conserved in the system?
Why is it that when both have speed v/√2, the relative speed of both masses moving is √2v?

Thank you.

 

[PeroK]

When one mass is held fixed, there an external force applied to prevent it from moving, but it is not applied over a distance - is this why energy is conserved in the system?
Why is it that when both have speed v/√2, the relative speed of both masses moving is √2v?

Yes, that's the basis of the argument. You can formalise it by integrating the force by $dr$ in both cases to derive the same PE for both systems. Hence the same KE at each separation.

 

[haruspex]

Why is it that when both have speed v/√2, the relative speed of both masses moving is √2v?

If each is moving at speed v/√2 towards the other then their closing speed is twice that, v√2.