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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)

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  • $\begingroup$ This sounds very wrong, if what you quote is correct. The center of mass tells us nothing about how each individual car moves. $\endgroup$ – Hanting Zhang Jan 17 at 19:33
  • $\begingroup$ Using the usual introductory textbook understanding of what a "collision" is (the bodies interact briefly with each other while in otherwise free motion), all collisions have the CoM of the participants in inertial motion, which means that there is a valid inertial frame in which that CoM is at rest. (Of course, the math of "collision"s is useful for non-free cases in cases where the participants are not free when the interaction forces are much larger than the ambient forces, so the intro book understand should be understood as an idealization.) $\endgroup$ – dmckee Jan 18 at 2:26
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You are correct, it is not guaranteed that the objects will come to a dead stop unless the collision is perfectly inelastic. If the objects are elastic or partially inelastic, they will have some motion after the collision that keeps the centre of mass where it was. The question assumes that collisions between cars are perfectly inelastic.

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You are correct. Given only the information in the question, there are other possibilities than just coming to a dead stop.

As an aside, as long as there are no external forces acting on a system, the center of mass will always move at a constant speed, regardless of the internal dynamics. This means that in any system with no external forces acting on it, you can always find a frame of reference* where the center of mass stays in exactly the same place. "Moving so that the center of mass of the two-car system stays in exactly the same place" is therefore not a particularly special or restrictive condition, because when the viewer is moving with the right velocity, that is true for any isolated system.

*Technically, this should be an inertial frame of reference, meaning that the viewer is moving with some constant velocity.

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