Verified answer

Forget about the spring! It's really just there to tell you that the initial momentum of the system is zero (neither cart is moving), and that the system is being acted upon by an internal force (so momentum will be conserved). Therefore,

0 = m1v1 + m2v2, or

v2 = -(m1/m2)*v1

The negative sign indicates that it moves in the opposite direction.

If m2 = 4*m1, then

v2 = - v1 / 4.

QED

BTW: the flaw in the logic of the other poster was in assuming that the spring decompressed from its center equally in both directions. It doesn't. (It probably decompresses from the center of gravity of the system, which is skewed toward the heavier mass, but that's just a guess.)

am i missing something here??

0.5 k x^2 = 0.5 m v^2

if they're connected to the same spring...then you can assume that they're both stretched to the equilibrium point of the spring (the middle)...so the compression distance is the same for both (x value)...its the same spring so k should be the same...

to solve for v...you get...

v = sqrt(k x^2 / m)

if cart A has 1/4 the mass of cart B...the sqrt makes it double the velocity...no??...i don't see how it can be four times as fast...