Don’t you need the distance between the two carts? 
Consider the time interval from when the two carts are released until
  just after they collide. For the system consisting of only the two
  carts, indicate whether the total mechanical energy increases,
  decreases, or remains the same. (The carts start out at rest and then stick in the collision btw)

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Don’t you need the distance between the two carts to even first calculate the Ug? 
I would say that Conservation of energy ( Ki+Ui=Kf+Ur) does not apply here because there is an unbalanced external force acting on the system of only the two carts; that force is the force of gravity due to the earth, which is very significant. 
But how would you know/prove that it increased or decreased? 
    Because technically, as the carts get closer, their U between each other decreases and their K increases, which makes it very fuzzy and intuitively ambiguous. 
****A lot of people misunderstand and say U=mgh. I know for sure this is NOT what this question asks though, because there is another of the exact same question with the two cart - AND earth system, where that U=mgh is true. But here it is regular Ug=Gm1m2/r equation I think. 
 A: The distinction your text makes, between a system consisting of only the two carts versus a system consisting of the two carts plus Earth's gravitational field, is not about imagining the two carts colliding due to their mutual gravitational attraction.  (Not least because that interaction is tiny; if they started a meter apart, total mass of a kilogram, it would take more than a day for them to touch.)
The distinction is an accounting difference: whether the work done by the Earth on the carts is an internal process, to be accounted as a change in potential energy, or an external process to be accounted as work done from outside.
A: The issues you are considering (distance between the carts and their possible gravitational interaction) are irrelevant. Internal conservative forces cannot change the total mechanical energy of the system, they can only cause conversions between kinetic and potential energy. The gravitational interaction between the carts is probably negligible, but that doesn't necessarily matter here.
Since the system is just the two carts, the gravitational force of the Earth on the system is an external force. i.e. we do not consider it in our potential energy. This means that we have an external force doing work on the system. I will leave it to you to conclude what this means for energy conservation of the system.
