I had the following problem:
Image two cars approaching one another in opposite directions. The first car is twice as massive as the second car. They move so that the center of mass of the two-car system stays in exactly the same place. What will happen when they collide?
[apologia Exploring Creation with Advanced Physics 1st Edition by Dr. Jay L. Wile. Question 1 of "Answers to the Review Questions for Module #5"]
The answer was that they would both come to a dead stop because the center of mass will continue to stay static.
Remember, any system of objects can be replaced by its center of mass. In this system, the center of mass is not moving. Thus, even after the collision, the center of mass will still not move. This means that once the cars collide, they will come to a dead stop.
[apologia Solutions and Tests for Exploring Creation with Advanced Physics 1st Edition by Dr. Jay L. Wile.]
However, since it didn't say that the collision was perfectly inelastic, couldn't the cars bounce backward at speeds such that their center of mass continues to remain in the same place? If not, why would they both come to a halt? Even using the answer key's explanation that the system of cars could be replaced by their center of mass, that still allows for the cars to rebound as long as the center of mass never moves.
(PS I am homeschooling which explains both my access to the answer key and lack of a teacher to which to ask this question)