The momentum of the two carts will be the same
Explanation:
First of all, we notice that the system consisting of the two carts + the spring is an isolated system: this means that there are no external forces acting on it.
As a result, the total momentum of the system must be conserved.
Since the spring is at rest, the total momentum of the system after the two carts are released is:
[tex]p_f = p_1 + p_2[/tex]
where
[tex]p_1[/tex] is the momentum of the 1st cart
[tex]p_2[/tex] is the momentum of the 2nd cart
The total momentum of the system at the beginning, when the spring is compressed, is zero:
[tex]p_i=0[/tex]
Since the total momentum must be conserved, we have:
[tex]p_i = p_f\\0 = p_1 + p_2\\\rightarrow p_2 = -p_1[/tex]
Which means that the two carts will have equal and opposite momenta.
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