In the movie there\'s a scene where two astronauts are joined together by a rope

Question

In the movie there's a scene where two astronauts are joined together by a rope. One is accelerating to pull the other. The astronaut would accelerate, the rope would tense and give a jerk to the other astronaut. Then the rope would loosen again until the accelerating astronaut gained enough velocity to tense the rope again, giving another jerk.

If something had very little mass, I believe the rope would simply remain tense. My question is, how do you determine if the rope remains tense or loosens? How much more velocity does the astronaut being pulled gain over the accelerating astronaut such that they loosen the rope?

Explanation / Answer

I haven't seen the movie. But if there are no external forces acting on either astronaut (there is just the tension in the rope), then by Newton's third law, the tension in the tether must result in equal and opposite forces (ie. the tension) being applied to both astronauts for the same time. So, if one astronaut pulls on the rope, absent any forces other than the pull on the rope, the two astronauts to move toward each other and keep moving together until they collide. When they collide, if one or both of the astronauts push off against the other, the two astronauts will move apart until the rope becomes tight and stops them.

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