Here is the question:

Two small identical smooth blocks $A$ and $B$ are free to slide on a frozen lake. They are joined together by a light elastic rope of length $L \sqrt(2)$ which has the property that it stretches very little when the rope becomes taut. At time t = 0, A is at rest at x = y = 0 and B is at x = L; y = 0 moving in the positive y-direction with speed V. Determine the positions and velocity of A and B at time

$t = 2L/V$

The hints and solutions are found here: https://dejanphysics.files.wordpress.com/2016/10/gnadig_1.pdf (under pdf pages 59 and 113)

I was confused because the hint seemed strange where it said that energy is conserved. Since the string becomes taut at some point, then doesn't some of the energy go into the spring potential energy of the rope, so not all of it is used on the blocks?

I assumed this, so I wrote out conservation of angular momentum about the CM of the system:

Before momentum = Momentum after:

$m(v)(L/2) = 2m(v_1)(\frac{L}{2\sqrt{2}})$ (*)

where v_1 is the velocity after they start both moving. Also, in the translational aspect, the CM moves with velocity $V/2$, so I was getting that right. But when I used formula (*) to find the angular velocity and positions of A,B and their velocities, I was getting very wrong answers. Can someone please explain where my mistake is?